Discussion Paper No. 792

AIRPORT PRIVATIZATION
AND INTERNATIONAL COMPETITION

Toshihiro Matsumura
Noriaki Matsushima

September 2010

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Airport privatization and international competition∗
Toshihiro Matsumura
Institute of Social Science, University of Tokyo
Noriaki Matsushima†
Institute of Social and Economic Research, Osaka University
September 11, 2010

Abstract
We provide a simple theoretical model to explain the mechanism whereby privatization of international airports can improve welfare. The model consists of adownstream (airline) duopoly with two inputs (landings at two airports) and two
types of consumers. The airline companies compete internationally. Using the simple international duopoly model, we show that the outcome where both airports
are privatized is always an equilibrium while that where no airport is privatized isanother equilibrium only if the degree of product differentiation is large.

JEL classification: L33, L13, R48
Key words: Airline, Airport, Privatization, International competition, Vertical relations∗

We would like to thank Masahide Daigo, Kojun Hamada, Toshio Matsuzawa, Kunio Miyashita,
Keizo Mizuno, Se-il Mun, Takeshi Nakata, Naoko Nishimura, Hisashi Oi, Kenichi Shoji, Koji Takamiya,Takeshi Yamazaki, Muneki Yokomi, and seminar participants at Niigata University, Shinshu University,and the Japan Society of Transportation Economics for their helpful comments. The authors gratefully
acknowledge financial support from Grant-in-Aid from the Japanese Ministry of Education, Science andCulture. Needless to say, we are responsible for any remaining errors.
†
Corresponding author: Noriaki Matsushima, Institute of Social and Economic Research, Osaka
University, Mihogaoka 6-1, Ibaraki, Osaka 567-0047, Japan. Phone: +81-6-6879-8571, Fax: +81-6-68798583, E-mail: nmatsush@iser.osaka-u.ac.jp1

1

Introduction

Following the example of the UK, many countries have moved—or are moving—towards
privatization of some public airports. Research on airport privatization has been popular recently (Basso (2008), Basso and Zhang (2008), Zhang and Zhang (2003, 2006)).
For instance, in their seminal paper on airport privatization, Zhang and Zhang (2003)
raise the following question: can we expect that the profit-maximizing behavior of privatized, unregulated airports will lead to social welfare maximization, or is there a conflict
between maximizing social welfare and maximizing profits? They incorporate the problem of congestion, capacity investment, and market power over landing charges into a
privatization policy framework and compare the price decisions of privatized, unregulated airports with those of public airports that maximize social welfare. Although thesestudies of airport privatization provide interesting insights, they do not show situations
in which privatization improves welfare; thus a welfare-maximizing government has no
incentive to privatize airports.
In this paper, we present a model in which local welfare-maximizing governments
have incentives to privatize airports and show why many countries have moved towards
privatization of public airports. We consider a simple downstream (airline) duopoly with
two inputs (landings at two airports). The model setting is as follows.1 There are two
downstream firms (airlines) that supply final products to consumers (passengers). Eachairline company procures two inputs for its final product. Each input is monopolisticallysupplied by an independent supplier. In the context of airline competition, the inputs
are related to airports commonly used by the airline companies.
We consider the following three-stage game. First, each government independently
decides whether to privatize its airport. The objective of a public airport is to maximize
domestic welfare, while that of the private airport is to maximize its own profit.2 In thesecond stage, the airports independently set their airport charges. In the third stage, two
1

We do not consider the problems of congestion and capacity at airports, although these are often

discussed in the papers mentioned above. Our purpose is to show that privatization can be beneficial
even though such negative effects, which might be mitigated by privatization, are absent.
2

This, in addition to the competition structure in the second and the third stages, follows the model

assumption in Zhang and Zhang (2003).

2

airlines face price competition and set their prices simultaneously.
We show that each government privatizes its airport so as to maximize its domesticsocial surplus in equilibrium. We also find that the governments may face a prisoners’
dilemma situation, that is, commitments to nonprivatization policies may be mutually
beneficial for the two countries.
The intuition behind the result is as follows. The strategic interaction between the
airports is that of strategic substitution. That is, when an airport charge is high (low), thecharge set by another airport becomes low (high). This is because a higher airport charge
diminishes the demand for each airline, which is closely correlated to the derived demand
for the other airport. Given the strategic interactions, we now consider the relation
between the airport charges and the objectives of the airports. A national airport sets alower airport charge than a privatized one because the national airport considers not only
its own profit but also that of the domestic airline company and consumer welfare. Whenthe other airport is foreign, this lower charge set by the national airport is the transferfrom it to the other airport. This causes a welfare loss that can dominate the welfare gain
for consumers and the profit of the domestic airline company. Because privatization ofan airport is a commitment not to lower its airport charge, this can improve its domesticsocial surplus.
The remainder of the paper is organized as follows. Section 2 presents the model.
Section 3 provides the main result. Finally, Section 4 concludes the paper.

2

The model

There are two downstream firms, D1 and D2 , which supply final products to consumers.
Each maximizes its own profit. The final products are differentiated. In the context ofairline competition, the downstream firms are associated with airline companies.
Each downstream firm procures two inputs (A and B) for its final product. Inputs
A and B are supplied by monopolistic common suppliers, Ui (i = A, B). In the context
of airline competition, the inputs are related to the use of airports. To produce a unit of
final product, each downstream firm uses one unit of input A and one of input B:
qi = min{qiA , qiB },3

where qij is the amount of input j (j = A, B). That is, each downstream firm has Leontief
production technology. Suppliers incur no marginal cost for a unit of input. Except for
procurement costs, the production costs of the downstream firms are zero. We assume
that the suppliers unilaterally set the input prices, wA and wB , respectively. We also
assume that the suppliers do not price discriminate. In this setting, per-unit cost of eachdownstream firm is given as
ci = wA + wB , (i = 1, 2).
There are two consumer groups, CA and CB , which differ in size. The size of group A
is λA = 1 and that of group B is λB = λ(≤ 1). That is, the size of group A is larger thanor equal to that of group B. The total demand for firm i’s product is given as qiA + λqiB .
Each consumer group j has the inverse demand functions of the products, denoted by(j = A, B)
pij = 1 − qij − γq−ij , i, −i = 1, 2, (i 6= −i),where qij is the amount of firm i’s product and γ is the degree of product differentiation(γ ∈ (0, 1)). From the inverse demand functions, the demand functions are derived
qij =1 − γ − pij + γp−ij, i, −i = 1, 2, (i 6= −i).
1 − γ2

When γ = 1, the products are perfect substitutes; when γ = 0, they are independent. In
the context of airline competition, the two airlines compete on the same route between
airports A and B because the two downstream firms use the common input suppliers(UA and UB ). Therefore, it is reasonable that the products are substitutes.
The profit of downstream firm i (i = 1, 2) is given as
πDi = (piA − (wA + wB ))qiA + (piB − (wA + wB ))λqiB , i = 1, 2.
The net consumer surplus in country j (j = A, B) is given as (λA = 1 and λB = λ)
Ã!
2 + 2γq q + q 2
q1j
1j 2j
2j
CSj = λj q1j + q2j −− p1j q1j − p2j q2j
2
Ã
!
µ
¶
2 + 2γq q + q 2
q1j2(1 − γ)(1 − p1j )(1 − p2j ) + (p1j − p2j )2
1j 2j
2j
= λj
.
= λj
2(1 − γ 2 )24The profit of upstream firm i (i = A, B) is given as
πU i = wi (q1A + q2A + λ(q1B + q2B )), i = A, B.
In the model, we assume that the nationalities of D1 , UA , and CA (D2 , UB , CB ) arecommon, and also assume that those of D1 and D2 differ.3 For instance, in the context
of airline competition, UA is a domestic airport from the viewpoint of D1 and UB is aforeign one. The market structure is summarized in Figure 1.
[Figure 1 here]
We consider two cases concerning the objective of each input supplier. One is thatthe objective of an input supplier is to maximize its own profit. The other is that theobjective of an input supplier is to maximize its domestic social surplus. In the context
of airline competition, the former relates to a privatized airport, and the latter to anationalized airport. This specification is used in the literature of airport privatization(e.g., Zhang and Zhang (2003), Basso and Zhang (2007)). Because two suppliers exist,
we must consider four cases concerning the objectives of the two suppliers: (1) Bothsuppliers are profit maximizers; (2) UA is a profit maximizer and UB is a domestic
welfare maximizer; (3) UB is a profit maximizer and UA is a domestic welfare maximizer;(4) Both suppliers are domestic welfare maximizers. When Ui is a profit maximizer, it
maximizes πU i (i = A, B). When UA (UB ) is a domestic welfare maximizer, it maximizes
πU A + CSA + πD1 (πU B + CSB + πD2 ).
The game proceeds as follows. First, each government simultaneously determines
whether or not it privatizes its input supplier. Second, each common supplier simultaneously sets wi to maximize its objective, which is determined in the first period (i = A, B).
Third, given the wholesale prices wA and wB , the downstream firms simultaneously set
their prices at pi (i = 1, 2). We assume that the downstream firms cannot price discriminate between the consumer groups A and B. That is, piA = piB = pi (i = 1, 2).

3

ResultWe solve the game by backward induction.
3

Of course, the nationalities of UA and UB (CA and CB ) are different.

5

3.1

The third stage

We first consider the third-stage game. The objective function of downstream i is:
πDi ≡ (pi − wA − wB )(qiA + λqiB )(pi − wA − wB )(1 + λ)(1 − γ − pi + γp−i )=, i = 1, 2, i 6= −i.
1 − γ2
The first-order conditions lead topi =1 − γ + wA + wB
(1 + λ)(1 − γ)(1 − wA − wB )2, πDi =
, i = 1, 2, j 6= i.
2−γ(2 − γ)2 (1 + γ)

(1)

The domestic social surplus in country A (B) is the sum of the consumer surplus incountry A (B) and the total profits of D1 and UA (D2 and UB ). The social surplus incountry j (j ∈ {A, B}) is given as
SWj =λj (1 − wA − wB )2(2 − γ)2 (1 + γ)
(1 + λ)(1 − γ)(1 − wA − wB )2 2wj (1 + γ)(1 − wA − wB )
+
+
, (2)
(2 − γ)2 (1 + γ)
(2 − γ)(1 + γ)

where λA = 1 and λB = λ.

3.2

The second stage

There are four cases (subgames) that depend on the first-stage decisions of the governments: (1) Both suppliers are profit maximizers; (2) UA is a profit maximizer and UB is
a domestic welfare maximizer; (3) UB is a profit maximizer and UA is a domestic welfare
maximizer; (4) Both suppliers are domestic welfare maximizers.
Privatization of the suppliers, UA and UB .

We first consider the case in which theobjective of the suppliers is to maximize their own profits. In other words, we consider
the situation after privatization of the suppliers, UA and UB .
The maximization problem of supplier i is represented by
max wi (q1A + q2A + λ(q1B + q2B )).
wi(3)

Let the superscript ‘PP’ denote the equilibrium value when the suppliers are private
firms. The maximization problems represented in (3) lead towi (w−i ) =1 − w−i21PPPP(i = 1, 2, −i 6= i) → wA
= wB
= .
3
6

(4)

This result is summarized in the following lemma:
Lemma 1 The strategic interaction between the suppliers is that of strategic substitution.
When the suppliers are profit maximizers, the equilibrium wholesale price in the subgame
P P = w P P = 1/3.
is wA
B

The reaction function in (4) means that the strategic interaction between the suppliers
is strategic substitution. When a supplier sets its wholesale price at a higher level, thetotal quantity demanded by the downstream firms shrinks. Given the shrunken demandcaused by the high price, the other supplier is forced to set a lower price (Cournot (1838)
and Sonnenschein (1968)). Note that this situation exists irrespective of the suppliers’
objectives (profit or welfare maximization).
The profits of the firms are given as
πUP iP =2(1 + λ)(1 − γ)(1 + λ)
PP
, πDj
=
, i = A, B; j = 1, 2.
9(2 − γ)(1 + γ)
9(2 − γ)2 (1 + γ)

The domestic social surplus in each country is given as
SWAP P =6 + 5λ − 3(1 + λ)γ
5 + 6λ − 3(1 + λ)γ, SWBP P =.
29(2 − γ) (1 + γ)
9(2 − γ)2 (1 + γ)

Only UA is privatized(5)

Second, we consider the case in which the objective of UA(UB ) is to maximize its own profit (its domestic social surplus). In other words, weconsider the situation in which only UA is privatized.
The maximization problem of supplier i is represented by
max wA (q1A + q2A + λ(q1B + q2B )),wA

max wB (q1A + q2A + λ(q1B + q2B ))

(6)

wB

+ (p2 − wA − wB )(q2A + λq2B ) +2 + 2γq q2
λ(q1B
1B 2B + q2B ).
2

Let the superscript ‘PN’ denote the equilibrium value when only UA is a private firm.
The maximization problems represented in (6) lead to
1 − wB
1 − wA, wB (wA ) =,2
3 + 2λ − (1 + λ)γ(2 − γ)(1 + λ)
1
PN
PN
wA
=
, wB
=
.
5 + 4λ − 2(1 + λ)γ
5 + 4λ − 2(1 + λ)γwA (wB ) =→P N > w P N . This is summarized in the following lemma:
We easily find that wA
B

7

Lemma 2 When only UA is privatized, it sets a higher wholesale price than UB . That
P N > wP N .
is, wAB

The reason is as follows. The domestic welfare maximizer (UB ) takes into account thedomestic consumer surplus and the domestic downstream firm’s profit as well as its own
profit. Because a lower wholesale price helps both consumers and the domestic firm, thedomestic welfare maximizer (UB ) sets a lower price than the profit maximizer (UA ).
The profits of the firms are given as2(2 − γ)(1 + λ)32(1 + λ)2
PN
,
π
=
,
U
B
(1 + γ)(5 + 4λ − 2(1 + λ)γ)2(1 + γ)(5 + 4λ − 2(1 + λ)γ)2(1 − γ)(1 + λ)3
PN
PN
πDA
= πDB
=
.
(1 + γ)(5 + 4λ − 2(1 + λ)γ)2
N
πUP A
=

The domestic social surplus in each country is given by
SWAP N =(6 + 5λ − 3(1 + λ)γ)(1 + λ)2(3 + 2λ − (1 + λ)γ)(1 + λ)2
PN
,
SW
=
.
B
(1 + γ)(5 + 4λ − 2(1 + λ)γ)2(1 + γ)(5 + 4λ − 2(1 + λ)γ)2Only UB is privatized(7)

Third, we consider the case in which the objective of UB (UA )is to maximize its own profit (its domestic social surplus). In other words, we consider
the situation in which only UB is privatized.
The maximization problem of supplier i is represented by
max wA (q1A + q2A + λ(q1B + q2B ))wA+ (p1 − wA − wB )(q1A + λq1B ) +2 + 2γq q
2
q1A
1A 2A + q2A
,
2

(8)

max wB (q1A + q2A + λ(q1B + q2B )).
wB

Let the superscript ‘NP’ denote the equilibrium value when only UB is a private firm.
The maximization problems represented in (8) lead to
NP
wA
=

λ
(2 − γ)(1 + λ)
NP
, wB
=
.
4 + 5λ − 2(1 + λ)γ
4 + 5λ − 2(1 + λ)γ

N P < w N P . This is summarized in the following lemma:
We easily find that wA
B

Lemma 3 When only UB is privatized, it sets a higher wholesale price than UA . That
N P < wN P .
is, wA
B

8

The reason is similar to that in the previous case.
The profits of the firms are given by
2λ(1 + λ)22(2 − γ)(1 + λ)3
PN
,
π
,
=
UB
(1 + γ)(4 + 5λ − 2(1 + λ)γ)2(1 + γ)(4 + 5λ − 2(1 + λ)γ)2(1 − γ)(1 + λ)3
NP
PN
πDA
= πDB
=
.
(1 + γ)(4 + 5λ − 2(1 + λ)γ)2πUNAP =The domestic social surplus in each country is given by
SWAN P =(5 + 6λ − 3(1 + λ)γ)(1 + λ)2(2 + 3λ − (1 + λ)γ)(1 + λ)2
NP
,
SW
=
.
B
(1 + γ)(4 + 5λ − 2(1 + λ)γ)2(1 + γ)(4 + 5λ − 2(1 + λ)γ)2No supplier is privatized(9)

Finally, we consider the case in which the objective of

the suppliers is to maximize their own social surplus. In other words, we consider thesituation in which no supplier is privatized.
The maximization problem of supplier i is represented by
max wA (q1A + q2A + λ(q1B + q2B ))

(10)

wA

+ (p1 − wA − wB )(q1A + λq1B ) +2 + 2γq q
2
q1A
1A 2A + q2A,2

max wB (q1A + q2A + λ(q1B + q2B ))

(11)

wB

+ (p2 − wA − wB )(q2A + λq2B ) +2 + 2γq q2
λ(q1B
1B 2B + q2B ).
2

Let the superscript ‘NN’ denote the equilibrium value when no supplier is privatized.
The maximization problems represented in (10) and (11) leads to
NN
wA
=

λ
1
NN
, wB
=
.
(3 − γ)(1 + λ)(3 − γ)(1 + λ)

(12)

N N and w N N is summarized in the following
The result for the difference between wA
Blemma:N N ≤ w N N , and theLemma 4 When the suppliers are domestic surplus maximizers, wA
B

equality holds only if λ = 1. That is, the wholesale price in the country with a larger
number of consumers is smaller than in the one with a smaller number of consumers.
The nationalized supplier considers consumer welfare and the domestic downstream firm’s
profit as well as its own profit. Because consumer welfare is more important in the country
N N < w N N in equilibrium.
with a larger number of consumers (country A), wA
B

9The profits of the firms are given by2λ2, πUNBN =,2(3 − γ) (1 + λ)(3 − γ)2 (1 + λ)(1 − γ)(1 + λ)
NN
NN
πDA
= πDB
=
.
(3 − γ)2 (1 + γ)πUNAN =The domestic social surplus in each country is given by
SWAN N =3.3

2 + 3λ − (1 + λ)γ
3 + 2λ − (1 + λ)γ, SWBN N =.
2(3 − γ) (1 + γ)
(3 − γ)2 (1 + γ)

(13)

Privatization policy: The first stage

We now discuss decisions regarding privatization; that is, the first stage of the game isdiscussed. The first stage is represented by the following 2 × 2 matrix.
A/B

P

N
SWBP P in (5)P
SWAP P in (5)SWBP N in (7)SWAP N in (7)SWBN P in (9)N
SWAN P in (9)SWBN N in (13)SWAN N in (13)

From the payoff matrix, we have the following proposition.
Proposition 1 For any exogenous parameter, (P, P ) is an equilibrium outcome. If thefollowing inequalities are satisfied, (N, N ) is an equilibrium outcome:p√−(1 − 3γ + γ 2 ) + 7 − 8γ + 2γ 2
3− 6
λ≥and γ ≤
' 0.275.
(3 − γ)(1 − γ)
2(14)Neither (P, N ) nor (N, P ) appears in equilibrium.
Proof See the Appendix.
The following figure indicates the area in which (N, N ) can be an equilibrium outcome.
[Figure 2 here]
Proposition 1 states that the outcome where both governments privatize their own suppliers (henceforth, we call this P P equilibrium) is always an equilibrium, while that where
no government privatizes its own supplier (henceforth, we call this N N equilibrium) can10fail to be an equilibrium. Figure 2 indicates that the area for the existence of N N equilibrium is small. Thus, we believe that P P equilibrium is more likely to occur than N N
equilibrium.4
We now explain the intuition behind the proposition. As explained above, because
the strategic interaction between the suppliers is strategic substitution, lowering wi leads
to higher w−i (i = 1, 2, i 6= −i). Because the supplier −i is a foreign firm, this priceshift caused by a lower wi is a transfer from the domestic to the foreign supplier. This
causes a welfare loss from the viewpoint of domestic welfare in country i. In any case,this welfare loss is important for each country. Therefore, each government decides toprivatize its own supplier when it anticipates that the other supplier is privatized.
In this setting, the degree of competitiveness (γ) also affects the welfare property
concerning the objectives of the suppliers. When γ is large (competition between thedownstream firms is severe), the negative effect of double marginalization is small because
downstream firms cannot exert their market power. The strategic interaction between
the suppliers is relatively important for each country. Moreover, the difference between
the number of consumers in the countries is also important. The supplier in the country
with a larger number of consumers sets its wholesale price at a lower level (Lemma 4).
This causes a transfer from this country to the foreign country. To reduce such a transfer,the supplier in the country with a larger number of consumers tends to be privatized.
We now briefly discuss the relation between the levels of social welfare in two cases:(1) both upstream suppliers are privatized, (P, P ); (2) no upstream supplier is privatized,(N, N ). When both upstream suppliers are privatized, the sum of wholesale prices wA +wB is higher than that in the case where no upstream supplier is privatized; that is,
P P + w P P > w N N + w N N . Although the higher wholesale prices in the privatized case
wA
B
A
B

reduce ‘global’ welfare, this negative effect of privatization is not always applied to thesocial surplus in the large country (country A). After a simple calculation, we have thefollowing proposition:

4

Even when two equilibriums exist, we can show that P P equilibrium risk dominates that of N N

equilibrium, so the former is more robust. For the concept of risk dominance, see Harsanyi and Selten(1988).

11

Proposition 2 SWBP P < SWBN N for all λ and γ. SWAP P > SWAN N if and only if
λ<3(2 − γ)(−3 + 6γ − 2γ 2 ).
(3 − γ)(21 − 22γ + 6γ 2 )Figure 3 indicates the area in which the inequality in Proposition 2 is satisfied.
[Figure 3 here]
From the figure, we can see that privatization of the two upstream suppliers improvessocial welfare in the large country when downstream competition is severe and the heterogeneity of the consumer sizes is significant.
N N < w N N . w N N in (12) is increasing in γ and decreasing
As shown in Lemma 4, wA
B
B

in λ. That is, when γ is large and λ is small, the rent shifting from country A to B issignificant. We now briefly explain the reason for this. When γ is large (downstream
competition is severe), the profits of the downstream firms are small. This implies that
consumer welfare and the upstream profits are more important than the downstream
profits. Therefore, each nationalized upstream supplier tends to set a higher price, whichharms the downstream firms. A higher wholesale price harms consumer welfare but
increases its upstream profit. Because the former negative effect is relatively important
for the large country, A, wA tends to be lower than wB . When λ is small (the difference
N N and w N N is large. Thisbetween the consumer sizes is large), the difference between wBA

is because the nationalized supplier in country A takes into account its (relatively) large
consumer size when it sets wA . Moreover, the strategic substitutability of the upstream
prices exacerbates the difference between the wholesale prices. Therefore, when γ is large
and λ is small, privatization of the two upstream suppliers improves social welfare in thelarge country, that is, SWAP P > SWAN N .

4Concluding remarks

We provide a simple theoretical model to explain the mechanism by which a government
privatizes its international airport to maximize domestic welfare. The model consists ofa simple downstream (airline) duopoly with two inputs (two airports). Using the simple
international duopoly model, we show that privatization of the airport can improve thedomestic social surplus. A lower price set by an airport leads to higher prices set by the12other airport. Because the other airport is a foreign one, this price shift is the transferfrom the domestic to the foreign one. This causes a welfare loss from the viewpoint ofdomestic welfare.
In this paper, we do not consider congestion problems, which are often discussed in
the literature on airline competition (Brueckner (2002, 2009), Yuen et al. (2008), Zhang
and Zhang (2006, 2010)). Considering this problem will make the analysis richer, andthis is a consideration for future research.

13

Appendix
Proof of Proposition 1:

SWAP P − SWAN P and SWBP P − SWBP N are given as

SWAP P − SWAN P=(2 − γ + (1 − γ)λ)(3(2 − γ)2 + 2(2 − γ)(7 − 3γ)λ + (17 − 14γ + 3γ 2 )λ2 )> 0,9(2 − γ)2 (1 + γ)(2(2 − γ) + (5 − 2γ)λ)2SWBP P − SWBP N=(1 − γ + (2 − γ)λ)((17 − 14γ + 3γ 2 ) + 2(2 − γ)(7 − 3γ)λ + 3(2 − γ)2 λ2 )> 0.
9(2 − γ)2 (1 + γ)(5 − 2γ + 2(2 − γ)λ)2(P, P ) is an equilibrium outcome for all of the parameters, γ ∈ (0, 1) and λ ∈ (0, 1]. These
inequalities imply that neither (N, P ) nor (P, N ) is an equilibrium outcome. SWAN N −SWAP N and SWBN N − SWBN P are given as
SWAN N − SWAP N=(2 − γ + (1 − γ)λ)(−(2 + 2γ − γ 2 ) + 2(1 − 3γ + γ 2 )λ + (3 − γ)(1 − γ)λ2 ),(3 − γ)2 (1 + γ)((5 − 2γ) + 2(2 − γ)λ)2SWBN N − SWBN P=(1 − γ + (2 − γ)λ)((3 − γ)(1 − γ) + 2(1 − 3γ + γ 2 )λ − (2 + 2γ − γ 2 )λ2 ).
(3 − γ)2 (1 + γ)(2(2 − γ) + (5 − 2γ)λ)2

SWAN N − SWAP N ≥ 0 (SWBN N − SWBN P ≥ 0) if and only ifp−(1 − 3γ + γ 2 ) + 7 − 8γ + 2γ 2,⇔λ≥(3 − γ)(1 − γ)p(1 − 3γ + γ 2 ) + 7 − 8γ + 2γ 2SWBN N − SWBN P ≥ 0 ⇔ λ ≤.
2 + 2γ − γ 2
SWAN N − SWAP N ≥ 0

After some calculus, we find that both inequalities are satisfied if and only if γ ≤ (3 −p√6)/2. We also easily show that (1 − 3γ + γ 2 + 7 − 8γ + 2γ 2 )/(2 + 2γ − γ 2 ) is larger√
than 1 if γ ≤ (3 − 6)/2. Therefore, the condition under which (N, N ) is an equilibriumoutcome is given byp√−(1 − 3γ + γ 2 ) + 7 − 8γ + 2γ 2
3− 6
λ≥and γ ≤
' 0.275.
(3 − γ)(1 − γ)2This calculus leads to Proposition 1.

Q.E.D.

14

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41(3), 1087-1119.
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UA

UB

wA
Route 1 (D1 )
CA A

wB

D1

D2

B CB
Consumers

Route 2 (D2 )Market structure

Vertical structure in this market.

Figure 1: The market structureλ1(N, N )(P, P )

1/2

(P, P )
γ

0

1/2

1

Figure 2: The condition that (N, N ) is an equilibrium outcome.

λ
0.4
0.3
0.2
0.1
0

SWAP P < SWAN N

SWAP P > SWAN N
1/2

γ
1

Figure 3: The condition that SWAP P > SWAN N .

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