Is competition in the transport industry bad? A welfare
analysis of R&D with inter-regional transportation
Kazuhiro Takauchi
Tomomichi Mizuno
July 2019
Discussion Paper No.1910

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANIs competition in the transport industry bad? A welfare
analysis of R&D with inter-regional transportation∗
Kazuhiro Takauchi†
Faculty of Business and Commerce, Kansai University
Tomomichi Mizuno‡
Graduate School of Economics, Kobe University
First version: August 29, 2018 This version: July 17, 2019
Abstract
We consider the welfare eﬀects of a higher number of carriers in a two-region reciprocal market
model. While exporting firms engage in cost-reducing R&D and use transport services to export
their products to the foreign market, carriers haul the exporting firms’ products and compete
in a Bertrand fashion in the inter-regional transport market. We demonstrate that if the degreeof R&D spillover is large, an increase in the number of carriers harms consumers and social
welfare. We further examine the case in which the exporting firm has monopsony power inthe transportation service, and also reveal that if the spillover rate is suﬃciently high, a higher
number of carriers reduces consumer and social surplus. Hence, policies to restrict competitionin the transport industry may be socially desirable.

Key words: Transport industry; Price competition; Cost-reducing R&D; Spillovers
JEL classification: L13; F12; R40∗
An earlier version of this paper was circulated with the title “Transport eﬃciency, downstream R&D, andspillovers.” We would like to thank Naoto Jinji, Makoto Okamura, Noriaki Matsushima, Eiichi Miyagawa, andKaz Miyagiwa for their helpful comments. We also thank seminar participants at the 2016 Japanese Economic
Association Spring meeting (Nagoya University) and the 77th annual meeting of the Japan Society of International
Economics (Kwansei Gakuin University). All remaining errors are our own.
†
Corresponding author: Kazuhiro Takauchi, Faculty of Business and Commerce, Kansai University, 3-3-35
Yamate-cho, Suita, Osaka 564-8680, Japan. E-mail: kazh.takauch@gmail.com; Tel.: +81-6-6368-1817; Fax: +816-6339-7704.
‡
Graduate School of Economics, Kobe University, 2-1 Rokkodai-cho, Nada-ku, Kobe-City, Hyogo 657-8501,
Japan. E-mail: mizuno@econ.kobe-u.ac.jp.

1

Introduction

The idea that “Promoting competition will benefit consumers” is widely believed among practitioners and competition authorities. For example, Joaquı́n Almunia1 stated: “[...] Our objective
is to ensure that consumers enjoy the benefits of competition, a wider choice of goods, of better
quality, and at lower prices. But competition not only delivers benefits for consumers, it alsodelivers benefits business and the economy as a whole.” (Almunia, 2010, p. 4). This idea isalso accepted with regard to industry such as ocean shipping, which has been strongly regulated
until now. Although, liner shipping industry has enjoyed an antitrust exemption until relatively
recently, the U.S. government reformed shipping act to promote competition and deregulationin the industry through narrowing the scope of antitrust exemption.2
However, there are conflicting views on the relationship between competition and consumer
benefit. While some empirical studies reveal that tougher competition due to an increased
number of firms lowers price and improves consumer surplus (e.g., Busso and Galiani, 2019;
Hausman and Leibtag, 2007),3 Other studies find that an entry can increase prices and may
possibly make consumers worse oﬀ (e.g., Caves et al., 1991; Grabowski and Vernon, 1992;
Thomadsen, 2007).4 Hence, it may be assumed that in reality, greater competition due to an
entry may not always be beneficial to consumers.
This paper presents a new perspective on the relationship between the promotion of compe-

1

Vice-President and Commissioner for Competition, European Commission 2010-2014.
In liner shipping industry, “conference” (also referred to as shipping conference) has traditionally been formedby shipping companies (i.e., carriers). The conference is a horizontal cartel that restricts competition among
carriers and engages in price-fixing, and it was immune from the application of the antitrust laws in many
developed nations (Stopford, 1988). However, from the mid-1980s, a competition promotion policy such as
independent fee-setting by conference members was introduced in the U.S., and the function of the conference
has therefore been weakened (Sagers, 2006).
3
Using a field experiment, Busso and Galiani (2019) examine the eﬀect of an entry of small retailers into amarket and they find that entry reduces prices. Hausman and Leibtag (2007) empirically demonstrate that WalMart’s entry lowered prices in traditional supermarkets and increased consumers’ benefit due to the increased
competition.
4
Caves et al. (1991) empirically demonstrate that market entry can lead to an increase in drug prices.
Grabowski and Vernon (1992) also find that an entry causes an increase in the price of pharmaceuticals. Using a simulation, Thomadsen (2007) demonstrates that duopoly prices can be higher than monopoly prices in a
first-food market.
2

1

tition and consumer welfare. By considering a market structure consisting of carriers (upstream
agents) and innovative exporting firms (downstream firms), we demonstrate that if the degreeof knowledge spillover among exporting firms is large enough, a higher number of carriers not
only harms consumers but also reduces social surplus.
We base our model on a Brander (1981) and Brander and Krugman (1983)-type reciprocal
market. While each region’s exporting firm uses inter-regional transport services and pays afreight charge to export to the foreign market, it can freely supply its local market. To reduce
their production costs, exporting firms engage in research and development (R&D) investment
involving knowledge spillovers.5 Inter-regional transportation is a homogeneous service, andcarriers compete on price à la Dastidar (1995).
We show that a higher number of carriers can reduce welfare. First, a rise in the numberof carriers reduces consumers surplus if and only if the degree of R&D spillover is large. A
higher number of carriers lowers transport prices, and its eﬀect becomes stronger as the spillover
rate increases. The transport price reduction increases the rival firm’s exports and decreases
domestic supply. Hence, when the transport price reduction begins to have a large impact on
the extent to which domestic supply decreases, total outputs fall because the decline in thedomestic supply exceeds the increase in the rival firm’s exports. Second, a higher number ofcarriers reduces social surplus in each region through a sharp decline in the domestic supply.
Exporting firms must pay a transport fee to export their products; thus, export activity is lesseﬃcient than supplying domestically. Although a higher number of carriers lowers transport
prices and promotes exports, it is equivalent to “fostering ineﬃcient production activity,” andit worsens production eﬃciency, which worsens social surplus.6
5This model setting is reasonable because previous studies empirically establish the existence of positive international spillovers from R&D. For example, see Coe and Helpman (1995) and Keller (1998).
6
In a diﬀerent context to our analysis, some studies consider the welfare eﬀects of the number of firms inan oligopoly market. Lahiri and Ono (1988) reveal that elimination of a high-cost firm (i.e., a firm with a low
market share) increases welfare. Dinda and Mukherjee (2014) demonstrate that when the government oﬀers
optimal uniform subsidy/tax, a higher number of high-cost firms reduces welfare. These studies focus on the cost
asymmetry among firms and they are quite diﬀerent from our model. Furthermore, Mizuno (2009) examines thewelfare eﬀects of a change in the number of upstream and downstream firms in a vertically related market with2

To avoid criticism of the assumption such that downstream firms are price takers in the
upstream market and also examine the robustness of our results, we further consider the casein which exporting firms have monopsony power regarding the transport service. The negativeeﬀect of an increased number of carriers on the domestic supply is moderated by this extension.
However, since the domestic supply drops sharply as the number of carriers increases if thespillover rate is suﬃciently high, a higher number of carriers can reduce the social surplus evenin this extended case.
Our study has an important policy implication. While competition authorities tend to
regulate horizontal mergers rigorously in the real world, our results indicate that reducing thenumber of carriers can raise both consumer and social surplus. Therefore, by facilitating mergers
in shipping industries through a revision of antitrust laws, competition authorities can increase
welfare. We believe that our model provides a new insight into the context of competition policy
and consumer protection.
This paper is most closely related to those of Takauchi (2015) and Takauchi and Mizuno(2019), who employ a similar market structure. In their inter-regional R&D rivalry setting,exporting firms must pay freight charges to ship their products to their rival’s domestic market,
but they freely supply to their local market. Takauchi (2015) examines the eﬀect of improving
the R&D eﬃciency on exporting firms’ profits. Takauchi and Mizuno (2019) consider a hold-up
problem resulting from carriers raising prices after observing an exporting firm’s investment,
and reveal that the solution to the hold-up problem, that is, adopting fixed-price contracts fortransport prices, can harm all firms. By contrast, we incorporate price competition among
carriers into a reciprocal market model and examine the eﬀects of the number of carriers on
welfare.
This paper is also related to research on the transport sector in the context of international
trade (Behrens et al., 2009; Behrens and Picard, 2011; Francois and Wooton, 2001; Ishikawa anddivisionalization.

3

Tarui, 2018).7 Francois and Wooton (2001) incorporate an imperfectly competitive transport
sector into a competitive trade model and examine the eﬀect of tariﬀ reductions. Behrens et al.
(2009) and Behrens and Picard (2011) examine the eﬀects of endogenous freight rates on the
agglomeration of firms. While Behrens et al. (2009) focus on the role of the carrier’s market
power, Behrens and Picard (2011) focus on the logistics problem associated with round trips.
Ishikawa and Tarui (2018) also examine the logistics problem and consider the role of trade
policies in oligopolies. While all of these studies use diﬀerent models to provide useful insights,
they do not consider the price competition among carriers and the R&D activities of exporting
firms.
In section 2, we present the baseline model and we derive our results in section 3. We extend
the baseline model to the case in which exporting firms have monopsony power regarding thetransportation service in Section 4. Concluding remarks are presented in section 5. We provide
all proofs in the appendix.

2

Model

We consider two regions, the H (Home) and F (Foreign) regions, whose product market issegmented from each other. Each region has an exporting firm, firm i (i = H, F ), that engages
in cost-reducing R&D activity and supplies its product to the local and other markets. The
inverse demand in region i is pi = a − Qi , where pi is the product price, Qi = qii + qji is total
output, qii is firm i’s domestic supply, qji is firm j’s exports, i, j = H, F , i ̸= j, and a > 0. The
region i’s consumers surplus is CSi = Q2i /2.
As firms have no means of carrying out long haul transportation, they pay a per-unit transport price, t, and use a transportation service to export their products. The profit of firm i
7

In addition, some studies treat transport prices (or costs) endogenously. Abe et al. (2014) examine tradeand environmental policies in a two-way duopoly in which transportation generates pollution. Matsushima and
Takauchi (2014) study port privatization and its eﬀects on the port usage fee (i.e., trade cost) in a two-way
oligopoly model.

4isΠi ≡ (pi − ci )qii + (pj − ci − t)qij − x2i for i ̸= j,where xi is firm i’s investment level and x2i is the R&D cost. Firm i’s production cost afterinvestment is ci ≡ c − xi − δxj ; that is, although firm i invests xi to reduce the unit cost c, there
is a knowledge spillover and the firm i enjoys some part of its rival’s developed knowledge, δxj ,without any payments. δ ∈ [0, 1] is the spillover rate of R&D and a − c > 0.
In the transport industry, there are n (≥ 2) identical cargo transporters, which we refer to
as carriers. For simplicity, we assume that carriers exist in regions besides the Home and Foreign regions. In our model, inter-regional transportation is a homogeneous service and carriers
compete in a Bertrand fashion. Let the transport price oﬀered by carrier k (∈ {1, . . . , n}) be
tk , carrier k’s individual transport demand be qk , and the aggregate demand be qHF + qF H .
Each firm employs the carriers oﬀering the lowest price, so the individual transport demand ofcarrier k is qk = [qHF (tl ) + qF H (tl )]/m if the carrier oﬀers the lowest price, tk = tl . Here, mdenotes the number of carriers oﬀering the lowest price. If carrier k oﬀers a slightly higher pricethan tl , then qk = 0. To obtain explicit solutions, we assume that carrier k has a quadratic
operation cost, (λ/2)qk2 , where λ > 0 denotes the transport eﬃciency.8 The profit of carrier k isπk ≡ tk qk − (λ/2)qk2 .
The timing of the game is as follows. In the first stage, each firm independently and simultaneously decides its investment level. In the second stage, the transport price is determinedthrough price competition among carriers. In the third stage, each firm independently andsimultaneously decides its level of its domestic supply and exports. The timing structure corresponds to the diﬃculty of a change in each decision. R&D generally takes much more time, so
its investment decision is in the first stage of the game. In contrast, since firms can frequently

8

The quadratic cost is popular in this type of price competition. For example, see Dastidar (1995 pp. 27),
Dastidar (2001 pp. 85), Delbono and Lambertini (2016a, 2016b), Gori et al. (2014), and Mizuno and Takauchi(2019).

5adjust their outputs, the production decision occurs in the last stage. Since in the second stage
the Nash equilibrium is not unique, we employ subgame perfect Nash equilibrium (SPNE) with
payoﬀ-dominance refinement as the equilibrium concept.9 We solve the game using backward
induction. We provide all proofs in Appendix B.

3

Results

In the third stage of the game, the first-order conditions (FOCs) to maximize the profit of firm
i are 0 = a − c − 2qii − qji + xi + δxj and 0 = a − c − qjj − 2qij + xi + δxj − t (i ̸= j). These
FOCs yield the following third-stage outputs of qii (t, x) = 13 [a − c + t + (2 − δ)xi + (2δ − 1)xj ]and qij (t, x) = 31 [a − c − 2t + (2 − δ)xi + (2δ − 1)xj ], where i, j = H, F , i ̸= j, and x = (xi , xj ).
In the second stage, the transport price t is determined by price competition among carriers.
As Dastidar (1995) demonstrates, if oligopolists with a convex cost engage in a homogeneous
price competition, the Nash equilibrium is not unique. In our model, the pure strategy Nashequilibria of transport price has a certain range of [ t, t̄ ] derived from the following two conditions:
The first condition is given by(πk (t, x, n) ≡ t

qHF (t, x) + qF H (t, x)
n

)

(
)
λ qHF (t, x) + qF H (t, x) 2
−
≥ 0,
2
n

which implies that “carriers do not raise their prices.” The second condition is given by
(
) λ(
)2
πk (t, x, n) ≥ πk (t, x, 1) ≡ t qHF (t, x) + qF H (t, x) − qHF (t, x) + qF H (t, x) ,2
which implies that “carriers do not lower their prices.” The first condition yields the lower
bound t, and the second yields the upper bound t̄:t=[2(a − c) + (xH + xF )(1 + δ)]λ(n + 1)[2(a − c) + (xH + xF )(1 + δ)]λ; t̄ =.
2(3n + 2λ)2[(3 + 2λ)n + 2λ]To narrow the equilibria, we employ the payoﬀ-dominance criterion that maximizes each9

For example, Cabon-Dhersin and Drouhin (2014) and Mizuno and Takauchi (2019) employ this concept.

6carrier’s profit. Since carriers are symmetric, the transport price is
tP = argmax πk (t, x) =t[2(a − c) + (xH + xF )(1 + δ)](3n + 4λ).
8(3n + 2λ)

The prices t, t̄, and tP ,10 yield Lemma 1.
Lemma 1 (i) tP > t. (ii) tP ≤ t̄ if and only if λ ≥ λ0 ≡ 3n/[2(n − 1)].
To ensure tP < t̄; that is, t = tP ,11 we require Assumption 1.
Assumption 1 λ > λ0 ≡ 3n/[2(n − 1)].

We next define z to facilitate the analysis.
Definition 1 z ≡ λ/n ∈ [3/2, ∞).12

Substituting the outcomes of the third and second stages into the profit of firm i and solving
the FOC, we obtain the following.
(a − c)[48z 2 + 144z + 113 − (4z + 5)(4z + 11)δ],E
8(a − c)(2z + 3)(4z + 5)
16(a − c)(2z + 3)∗
qii∗ =; qij=, i ̸= j,EE
8(a − c)(2z + 3)(4z + 3)
t∗ =
,
E
x∗i =

(1)
(2)
(3)

where E ≡ (4z + 5)(4z + 11)δ 2 − 2(16z 2 + 40z + 29)δ + 5(4z + 5)(4z + 7) > 0. The variable “∗”is the SPNE outcome.
The profits of carrier k and firm i are πk∗ =( 2z+3 )n∗ )2 and Π∗ = (q ∗ )2 + (q ∗ )2 − (x∗ )2 ,
(qij
i
ii
ij
i

respectively.
10

The price tP is also known as collusive-price because it maximizes joint profit of carriers. To narrow the setof Nash equilibria, this collusive-price refinement is often employed. For example, see Dastidar (2001), Gori etal. (2014), and Mizuno and Takauchi (2019). The collusive-price refinement is identical to the payoﬀ-dominancerefinement if the Nash equilibria contain the interior maximizing point of each carrier’s profit (i.e., the upperbound of the equilibria is strictly larger than the interior maximizer of each carrier’s profit).
11“t = tP ” is partially consistent with the characteristics of the transport industry. For example, as indicatedby Hummels et al. (2009), ocean shipping is an oligopoly market. Moreover, some studies report collusion in thisindustry (e.g., Sjostrom, 2004; Sys, 2009; Sys et al., 2011). Among them, Sys (2009) empirically demonstrates
that the containerized shipping industry is tacitly collusive.
12
Although we need z > (3/2)(1/(n − 1)) from Assumption 1 because the maximum of 1/(n − 1) is 1, λ > λ0
holds for all z ≥ 3/2.

7

To ensure a positive unit production cost after investment, we require Assumption 2.
Assumption 2 c/(a − c) > (1 + δ)[48z 2 + 144z + 113 − (4z + 5)(4z + 11)δ]/E.

From (1)–(3), we establish Lemma 2.
16z +40z+29∗ /∂δ < (=, >) 0.
Lemma 2 I. If δ > (=, <) δt ≡ (4z+5)(4z+11), ∂t∗ /∂δ, ∂qii∗ /∂δ, and ∂qij
2

II. (i) Suppose z < z1 ≃ 5.90928; then, ∂x∗i /∂δ < 0. (ii) Suppose z > z1 ; then, if δ < δx ,∂x∗i /∂δ > 0. Otherwise, ∂x∗i /∂δ ≤ 0. (The threshold δx is defined in Appendix B1.)
Similarly, (1)–(3) yield the following result.

Proposition 1 (i) Keener competition in the transport industry (i.e., a rise in n) and higher
transport eﬃciency (i.e., a fall in λ) decreases transport prices and domestic supply, but theseincrease exports. (ii) Keener competition in the transport industry and higher transport eﬃciency
increases the firm’s investment if and only if δ < 5/(8z + 7).

We first consider part (i) of Proposition 1. A higher n and a smaller λ (i.e., a decrease inz) have a similar eﬀect. A higher n lowers transport prices by intensifying competition among
carriers and it therefore increases exports. A smaller λ flattens the slope of the carriers’ cost
curve, which induces a lower transport price, and thereby promotes exports. Because both ahigher n and a lower λ expand imports and makes competition in the local market tougher, firm
i’s domestic supply falls.
Second, we examine the logic behind Lemma 2. A higher δ lowers production costs, facilitates
production activities, and thus increases both domestic supply and exports. That is, in ourmodel, δ has exactly the same eﬀect on both domestic supply and exports. Since a higher
δ leads to an expansion in transport demand, it encourages carriers to set higher prices. (Alower δ yields the inverse result.) Hence, if a higher (lower) δ increases (decreases) transportdemand, both outputs and transport prices increase (decrease) as δ goes up (down) because8it also raises (lowers) transport price. On the one hand, an increase in transport prices raises
the trade barrier, which impedes exports. If δ rises when its level is low enough, because thetransport price is low and the positive eﬀects of a reduction in production costs exceeds theexport impeding eﬀect of rising transport prices, the firm’s exports increase. Conversely, whenboth δ and transport prices are high, a rise in δ reduces exports because the export impeding
eﬀect becomes large. Transport demand then falls and carriers lower their prices as δ rises.13
A rise in δ has positive and negative eﬀects on the R&D motive. A higher δ encourages
investment because it reduces the unit production cost and facilitates production (positive effect). If δ increases, because each firm enjoys its rivals developed knowledge without cost, theR&D motive weakens (negative eﬀect). Investment usually decreases as δ rises because the negative eﬀect is dominant. This is a well-known result illustrated by d’Aspremont and Jacquemin(1988).
Diﬀerent from the standard result, in our model, the positive eﬀect can be dominant. Whenλ is large, that is, z is large, transportation is ineﬃcient and its price is high. A high transport
price impedes cross-hauling and strengthens the monopolization of the local firm in its market.
Suppose that the R&D spillover arises, that is, δ slightly increases from 0 in this case; theunit production cost then falls and outputs increase, but it also raises transport prices andthe domestic supply increases more rapidly than exports. This strengthens the degree of thelocal firm’s relative monopoly in its market. Because such enlargement in the domestic supply
promotes investment and the positive eﬀect becomes dominant, R&D investment increases as δ
rises. However, if δ goes above a certain level, the negative eﬀect is dominant since an inflow of
the rival firm’s developed knowledge becomes large.
We next consider part (ii) of Proposition 1, which indicates that to raise investment byAdditionally, ∂x∗i /∂δ can explain why the transport prices and the firm’s outputs have the same change forδ. From the third-stage outputs and t = tP , noting that xi = xj in[ equilibrium, the] total diﬀerentiation
[ of
5n+4λ
3n+4λ
i
xi + (1 + δ) dx
,
dq
/dδ
=
xi +
qii = qii (x, t, δ), qij = qij (x, t, δ), and t = t(x, δ) yields dt/dδ = 4(3n+2λ)iidδ4(3n+2λ)
]
[
]
dxi
n
i
(1 + δ) dx
,
and
dq
/dδ
=
x
+
(1
+
δ)
.
ij
i
dδ
2(3n+2λ)dδ
139lowering transport prices, δ should be suﬃciently small. The intuition is as follows. As weshow in Lemma 2, when δ is suﬃciently small, the transport price is also low and the degreeof export restriction is weak. Although a higher n and a smaller λ commonly reduce transport
prices and increase exports, they decrease the domestic supply. If δ is suﬃciently small, because
the transport price reduction leads to an increase in exports in excess of the decrease in thedomestic supply, a higher n and a smaller λ can encourage investment. In contrast, if δ is large,
a higher n and a smaller λ can never encourage investment because the degree of the reduction
in domestic supply becomes larger.
Does an increase in the number of carriers and improved transport eﬃciency enhance welfare?
We next focus on the welfare eﬀects of n and λ. To examine their eﬀects on consumer benefit,
we use (2) and obtain the following result.

Proposition 2 Keener competition in the transport industry and higher transport eﬃciency
reduces consumer welfare if and only if the R&D spillover rate is suﬃciently high; that is,∂Q∗i /∂z > 0 if and only if δ > δcs . (The threshold δcs is defined in Appendix B1.)
[Fig. 1 around here]

Panel (a) of Fig. 1 depicts Proposition 2. As long as the R&D spillover is not too high,the trade promotion due to the transport price reduction increases total output, Qi = qii + qji ,and lowers the product price and consumer surplus therefore increases. However, Proposition
2 indicates that this promotion of inter-regional trade is not always desirable for consumers. A
key to this result is the role of ∂t∗ /∂z.
A higher δ lowers the production costs of firms and increases aggregate transport demand.
When the aggregate transport demand is high, each carrier’s individual demand is also highbecause carriers are symmetric. Then, suppose that the number of carriers n increases; that is,z (≡ λ/n) decreases. Tougher competition among them reduces each carrier’s demand, and the10

size of the demand that the carrier loses increases as the aggregate transport demand increases.
Although each carrier lowers its price if there is a reduction in its demand, carriers sharply lower
their prices compared to the case of low aggregate transport demand because the size of the lostdemand increases as the aggregate transport demand increases. Hence, a higher δ strengthens∂t∗ /∂z.14
Although a higher δ facilitates production, it can raise transport prices (Lemma 2). A higher∗ /∂zδ has both export promotion and restriction eﬀects and a higher δ therefore strengthens ∂qji
∗ /∂z in the other cases. On the one hand, as we show inin some cases, but it weakens ∂qji

Proposition 1, a higher n (or lower z) reduces domestic supply because transport prices (i.e.,
the rival firm’s trade barrier) fall. If δ is high, the degree of reduction in domestic supply for an
increase in n is also large because the degree of the reduction in transport prices for an increasein n is large. That is, when δ rises, the “∂t∗ /∂z” eﬀect becomes stronger, which strengthens the“∂qii∗ /∂z” eﬀect.
∗ /∂z) as functions of δ in Panel (b) of Fig. 1. (Since ∂q ∗ /∂z
We illustrate ∂qii∗ /∂z and −(∂qjiji

has a negative value, we multiply it by −1.) As δ increases above a certain level, ∂qii∗ /∂z (the∗ /∂z) (the inverted U-shaped curve). Hence, if δ is suﬃciently
increasing curve) exceeds −(∂qji

high, ∂Q∗i /∂z has a positive value. When δ is high, a higher n (lower z) reduces total output.
z (≡ λ/n) and δ have the following eﬀects on the profits of firms and carriers.
I. ∂Π∗i /∂z > 0 and ∂Π∗i /∂δ > 0.

Lemma 3
∂

(∑ n

)

∗
k=1 πk /∂z < 0.

II. (i) ∂πk∗ /∂n < 0, ∂πk∗ /∂λ < 0, and16λ +40λn+29n, ∂πk∗ /∂δ < (=, >) 0.
(ii) If δ > (=, <) δt ≡ (4λ+5n)(4λ+11n)22

Lemma 3 is intuitive. Since a larger n and a smaller λ promote less eﬃcient production activity, that is, exports, they reduce the firm’s profit. In contrast, a higher δ decreases production
costs, and thus increases the firm’s profit. The carrier’s profit depends on the transport price,so ∂πk∗ /∂n and ∂πk∗ /∂δ correspond to the changes in the transport price. A smaller λ implies an
14∀z ≥ 3/2, (∂/∂δ)(∂t∗ /∂z) > 0 holds.

11

eﬃciency improvement in transportation, and it hence increases the carrier’s profit. A larger n,that is, fall in z, makes competition among carriers tougher and reduces each existing carrier’s
profit. However, the aggregate number of carriers increases and the total amount of their profits
increases.
Subsequently, we examine the eﬀects of a change in the number of carriers on each region’ssocial surplus and the entire world. The social surplus in region i (i = H, F ) is defined by thesum of consumers surplus and profit of firm i, which is given by(a−c)2SW ∗i =
E2[]

3840z 4 + 22016z 3 + 48160z 2 + 47712z + 18047.

+ 2(4z+5)(4z+11)(48z 2 +144z+113)δ − (4z+5)2 (4z+11)2 δ 2(4)World welfare, W W , is comprised of two countries’ social surplus and all profits of carriers,that is, W W ∗ ≡∑∗i=H,F SWi +2(a−c)2
WW∗ =
E2

[

∑n

∗
∗
k=1 πk . The πk and (4) yield]

3840z 4 + 23040z 3 + 52768z 2 + 54624z + 21503
2+ (4z+5)(4z+11)(48z 2 +144z+113)δ − (16z 2 +64z+55) δ 2.

(5)From (4) and (5), we establish Proposition 3.

Proposition 3 Suppose that all carriers belong to other than both two regions H and F . Then,
keener competition in the transport industry and higher transport eﬃciency (i) always reduceeach region’s social surplus (i.e., ∂SWi∗ /∂z > 0); (ii) reduce the world welfare if and only if
the R&D spillover rate is suﬃciently high, that is, ∂W W ∗ /∂z > 0 if and only if δ > δww . (Thethreshold δww ∈ (0, 1) is defined in Appendix B1.)

We first consider part (i). From the definition of social surplus, the symmetric outcomes(xi = xj , qii = qjj , qij = qji , and pi = pj ), and p′i = −1, we can decompose the eﬀects of a12change in z (e.g., a change in n or λ) on welfare as follows:
∂SWi
∂qii
= [pi − (c − (1 + δ)xi )]
+
∂z
∂z}
|
{z
(i) Domestic supply eﬀect (+)∂xi[(1 + δ)Qi − 2xi ]∂z}|{z(ii) Investment eﬀect (+)/(−)∂qij+ [pj − (c − (1 + δ)xi ) − t]
+
∂z}
|
{z
(iii) Export eﬀect (−)

(
)
∂t
−qij
∂z
|
{z
}

(6)
.

(iv) Transport price eﬀect (−)

There are four eﬀects (terms (i)–(iv)) in (6).15 The domestic supply eﬀect, term (i), is
positive because a decrease in z (i.e., an increase in n and a decrease in λ) raises the rival firm’s
exports and makes the competition in the domestic market tougher. The investment eﬀect,
term (ii), depends on a change in investment ∂x∗i /∂z and the eﬀect can therefore be positive
when the R&D spillover is low; otherwise, it is negative. Both the export and transport priceeﬀects, which correspond to terms (iii) and (iv), respectively, are negative. Since a decrease in z
promotes exports, the export eﬀect is negative. A fall in transport prices curtails the ineﬃciencyof cross-hauling; this eﬀect is therefore also negative.
In our model, the domestic supply eﬀect (term (i)) dominates any other eﬀects because theproduction of domestic supply is more eﬃcient than export activities. The diﬀerence in supply
eﬃciencies makes the domestic supply larger than that of exports, and hence, the domestic
supply sharply decreases when competition in the local market increases due to the promotionof the rival firm’s exports. A fall in z, that is, a rise in n, reduces welfare.
We illustrate this result in Fig. 2. If the spillover rate of R&D is low, the investment eﬀect
can be negative. Thus, all terms can be negative except for (i) if the spillover rate is low. Fig.
2 depicts each term when δ = 1/4. Even in such a case, we can immediately see that (i) (thecurve (i) in Fig. 2) is extremely large compared with the other terms.

[Fig. 2 around here]

In contrast, the industry profit of the transport sector is added in the world welfare and15

For more detail, see Appendix A2.

13

it may therefore be improved due to competition among carriers. As shown in Lemma 3, theindustry profit of the transport sector,∑n∗k=1 πi , increases as z decreases.

To understand part(ii), it is enough to incorporate this negative eﬀect into the argument of part (i). As illustrated
in the logic behind Proposition 2, a higher (lower) spillover rate strengthens (weakens) the eﬀectof a change in the domestic supply due to a change in z (i.e., ∂qii∗ /∂z). Hence, when the spillover
rate is small, the eﬀect of term (i) in (6) is not suﬃciently strong. Then, because the eﬀect of∑“∂( nk=1 πi∗ )/∂z” and other negative eﬀects (terms (iii) and (iv) in (6)) can be dominant, a fallin z improves the world welfare.
There is an important policy implication. Suppose that the spillover rate of R&D is suﬃciently high. Then, tougher competition due to an increase in the number of carriers worsens
consumers surplus, each region’s social surplus, and world welfare (Propositions 2 and 3). Thisimplies that promotion of competition in the transport industry through strengthening of mergers regulation or lowering barriers to entry can work negatively for economies. It may be desirable
for the competition authorities to recommend horizontal mergers within the transport industry
or to raise barriers to entry in that industry.

4

An Extension

Duopsony in the transport market
In Section 3, we assumed that firms are price takers in relation to the transport service, that is,
firms have monopoly power in the downstream product market, whereas they do not have monopsony power in the upstream transport market. Although this assumption is frequently employedin the study of vertically related markets, there is also a criticism that “while downstream firms
recognize their monopoly power and strategically behave as sellers in the downstream market,
they do not strategically behave as buyers and are price takers in the upstream market.”16 To
16
For this criticism see, for example, Ishikawa and Spencer (1999). They oﬀer some arguments to justify theassumption such that downstream firms are price takers in the upstream market.

14

avoid such criticism, we further examine the situation in which firms have monopsony power inthe transport service market.
To examine the situation in which the export decision of each firm directly aﬀects to thetransport supply, we consider the following timing of the game.
• First stage: The firm i (i = H, F ) chooses its investment level, xi .
• Second stage: Given the transport price t, each carrier k (k = 1, ..., n) chooses its freight
traﬃc, qk . Then, the shape of the inverse transport supply function, t = T (qHF + qF H ),is fixed.
• Third stage: Given the inverse transport supply function, firm i decides its exports, qij ,and domestic supply qii (i, j = H, F and i ̸= j).
• Fourth stage: The transport market is cleared by the equilibrium price, t.
The game is solved using backward induction. In Appendix A, we report the detailed procedureused to obtain the SPNE of this game and the necessary equilibrium outcomes.
When firms have monopsony power regarding the transport service, they can lower thetransport price by decreasing their export volume because their exports (i.e., volume of traﬃc)
influences to the inverse transport supply . Hence, in the duopsony case, the equilibrium transport price, td∗ , is lower compared to the case in which they are price takers, that is, t∗ > td∗ .17
From (A1), we establish the following proposition.
d and z > z ], keener competition in the
Proposition 4 (i) If z < z2 ≃ 2.58114 or [δ < δcs2transport industry and higher transport eﬃciency increase consumer welfare.
d and z > z , keener competition in the transport industry and higher transport(ii) If δ > δcs
2d > 0 is defined in Appendix B1.)eﬃciency reduce consumer welfare. (The threshold δcs

17

We summarize this result as “Lemma 4” in Supplementary Material.

15

[Fig. 3 around here]

A rise in the spillover rate, δ, strengthens the degree of a change in the transport price due toa change in z (i.e., the “∂t/∂z” eﬀect), and also strengthens the degree of change in the domestic
supply due to a change in z (i.e., the “∂qii /∂z” eﬀect). Hence, when δ is suﬃciently high, the“∂qii /∂z” eﬀect is dominant, and thus, the area such that ∂Q/∂z > 0 appears (see Fig. 1). On
the one hand, in the duopsony case, the equilibrium transport price is lower than that in theprice-taker case. Because the decline in the transport price makes the “∂t/∂z” eﬀect weaker,
the “∂t/∂z” eﬀect in the duopsony case becomes weaker than that in the price-taker case.18 As
shown in the logic behind Proposition 2, if the “∂t/∂z” eﬀect becomes weaker, the “∂qii /∂z”
eﬀect also becomes weaker. Hence, in the duopsony case, the “∂qii /∂z” eﬀect is weaker than thatin the price-taker case. Therefore, in the duopsony, the value of the spillover rate that makesd ) is higher than that in the price-taker case.
the “∂qii /∂z” eﬀect dominant (i.e., the threshold δcs

Panels (a) and (b) in Fig. 3 illustrate this relationship.
In addition, from (A2), we establish the following proposition.

Proposition 5 Suppose that all carriers belong to other than both two regions H and F . Then,
keener competition in the transport industry and higher transport eﬃciency reduce each region’ssocial surplus if and only if the R&D spillover rate is suﬃciently high. That is, ∂SWid∗ /∂z > 0
d . (The threshold δ d ∈ (0, 1) is defined in Appendix B1.)
if and only if δ > δsw
sw

The logic behind Proposition 5 is intuitive. A fall in z reduces the transport price andpromotes less eﬃcient production activity, that is, exports, and it therefore decreases the profit
of firms, as is the case for price takers.19 However, as shown in Proposition 4, the area in which
the total output (or consumers surplus) rises due to a decline in z expands, and hence, thearea in which the social surplus increases due to a decline in z can appear. Diﬀerent from the18
19

Using Mathematica plotting, we find that ∂t∗ /∂z > ∂td∗ /∂z > 0.
This result is reported in Supplementary Material.

16

price-taker case, in a duopsony case, even if there is no carrier within the region, an increase in
the number of carriers can enhance the social surplus of that region.

5

Conclusion

Many argue that promoting competition, such as by increasing the number of firms in a market,
weakens each firm’s market power and lowers prices, which makes consumers better oﬀ. However,this standard perspective on the relationship between competition and consumer welfare is notnecessarily robust and may not hold when we consider the inter-regional transportation.
We demonstrate that in a market structure consisting of carriers (or shipping firms) andinnovative exporting firms, a higher number of carriers sharply reduces the exporting firms’
domestic supply and therefore reduces total outputs and consumer surplus if the degree of
R&D spillover is large. We also reveal that a higher number of carriers promotes less eﬃcient
production activity, that is, exports, which can reduce social surplus.
We further extend our model to the case in which exporting firm has monopsony power
regarding the transportation service. This extension moderates the negative eﬀect of domestic
supply reduction due to an increased competition. On the other hand, because a higher numberof carriers sharply reduces the domestic supply when the rate of R&D spillover is suﬃciently
high, a higher number of carriers can also decrease consumer surplus in a duopsony case.
We do not consider diﬀerent forms of innovation activities among exporting firms, such as
product R&D and quality improving innovation. Although whether our main findings hold may
be interesting when exporting firms conduct these other R&D activities, this aspect is beyond
the scope of our analysis. It may be fruitful for future research to examine this relationship.

17

Appendix
A. The SPNE outcomes in duopsony of exporting firms
We present the calculation to derive the equilibrium outcomes in the game in which firms havemonopsony power in the transport market.
In the fourth stage of the game, the equilibrium transport price is decided so as to equalizetransport supply with its demand. However, the transport demand, total exports of two firms,
is chosen in the third stage. Thus, to solve the game correctly, we assume an inverse transportsupply function, t = T (qHF + qF H ) and consider this in the third stage.
• The third stage. From the profit of firm i and t = T (qHF + qF H ), the FOCs for profit
maximization of firms are ∂Πi /∂qii = 0 ⇔ a − c − 2qii − qji + xi + δxj = 0 and ∂Πi /∂qij = 0 ⇔a − c − 2qij − qjj + xi + δxj − t − T ′ (qHF + qF H )qij = 0 (i ̸= j). Let “′” be the first derivative andT ′ = T ′ ( · ). The FOCs yield the third-stage outputs: qii (t, x; T ′ ) = [a−c+(2−δ)xi +(2δ −1)xj +t + (a − c + xi + δxj )T ′ ]/(3 + 2T ′ ) and qij (t, x; T ′ ) = [a − c + (2 − δ)xi + (2δ − 1)xj − 2t]/(3 + 2T ′ ).
• The second stage. The carrier k’s maximization problem, maxqk πk , yields qk = t/λ.
Because the transport demand is qHF + qF H , the market clearing condition is qHF + qF H =∑n

k=1 qk = nt/λ. From this, the inverse transport supply in this subgame is t = T (qHF +qF H ) =(qHF + qF H )λ/n, and hence, T ′ = λ/n holds. Plugging t = (qHF + qF H )λ/n and T ′ = λ/n into
the third-stage outputs and solving these for outputs again, we obtain the second-stage outputs:qii (x) =(a − c)(3n + 2λ)(n + 3λ) + ui xi + uj xj,
3(n + 2λ)(3n + 2λ)qij (x) =n[(a − c)(3n + 2λ) + vi xi + vj xj ],3(n + 2λ)(3n + 2λ)where ui ≡ 2(3n2 + 8nλ + 3λ2 ) − n(3n + 5λ)δ, uj ≡ 2(3n2 + 8nλ + 3λ2 )δ − n(3n + 5λ), vi ≡2(3n + 5λ) − (3n + 8λ)δ, and vj ≡ 2(3n + 5λ)δ − (3n + 8λ).
H +xF ))λThe above qij (x) yields the second-stage transport price: t(x) = (2(a−c)+(1+δ)(x.
3(n+2λ)• The first stage. In this stage, each firm decides its investment level, xi . The objective

18

function of firm i, Πi (x), is derived from qii (x), qij (x), and t(x) . Solving the FOCs, ∂Πi (x)/∂xi =0 (i = H, F ), with respect to xi , we obtain the following SPNE investment level:xd∗i =(a − c)[2(9z 3 +32z 2 +25z+6)−(23z 2 +25z+6)δ]
,
K

where
K ≡ 54z 3 +116z 2 +76z+15 − (2z+3)(9z 2 +7z+2)δ + (23z 2 +25z+6)δ 2 > 0.

The SNPE outputs and transport price are3(a−c)(2z+1)(2z+3)(3z+1)
3(a−c)(2z+1)(2z+3)
d∗
; qij
=
,
K
K
6(a − c)z(2z + 1)(2z + 3)
td∗ =
.
K

qiid∗ =d∗ and Πd∗ = (q d∗ )2 + (q d∗ )2 − (xd∗ )2 .
The profit of carrier k and firm i are πkd∗ = n1 td∗ qij
i
ii
ij
i

The equilibrium outputs yield the total output in region i:
d∗
d∗
Qd∗
i = qii + qji =3(a − c)(2z + 1)(2z + 3)(3z + 2)
, j ̸= i.
K

(A1)

2
d∗
The social surplus in region i, SWid∗ = 21 (Qd∗i ) + Πi , is(a−c)2
SW d∗
=
i
2K 2

[

3240z 6 +14688z 5 +25720z 4 +23200z 3 +11683z 2 +3162z+360
2+ 8(23z 2 +25z+6)(9z 3 +32z 2 +25z+6)δ − 2(23z 2 +25z+6) δ 2

The world welfare, W W d∗ =WWd∗(a−c)2
=
K2

]

[

∑

d∗
i=H,F SWi +

(A2)

∑n

d∗
k=1 πk , is3240z 6 +14976z 5 +26872z 4 +24784z 3 +12547z 2 +3324z+360+ 8(23z 2 +25z+6)(9z 3 +32z 2 +25z+6)δ−2(23z 2 +25z+6)2 δ 2

B. Proofs and supporting calculations
B1. Proofs
H +xF )(1+δ)]Proof of Lemma 1. (i) A simple algebra yields tP − t = 3n[2(a−c)+(x> 0.
8(3n+2λ)3n
H +xF )(1+δ)][2λ(n−1)−3n], tP ≤ t̄ iﬀ λ ≥ 2(n−1). Q.E.D.
(ii) Since t̄ − tP = 3n[2(a−c)+(x8(3n+2λ)[(3+2λ)n+2λ]

19

.

]
.

Proof of Lemma 2.

I. Diﬀerentiating (2) and (3) with respect to δ yields ∂t∗ /∂δ = (16(a −∗ /∂δ = (32(a −c)(2z + 3)(4z + 3)/E 2 )L1 , ∂qii∗ /∂δ = (16(a − c)(2z + 3)(4z + 5)/E 2 )L1 , and ∂qij

c)(2z + 3)/E 2 )L1 , where L1 ≡ 16z 2 + 40z + 29 − (4z + 5)(4z + 11)δ. These yield part I.
II. Diﬀerentiating (1) with respect to δ yields ∂x∗i /∂δ = ((a − c)/E 2 )L2 , where L2 ≡ 256z 4 −512z 3 − 4640z 2 − 7008z − 3071 − 2δ(4z + 5)(4z + 11)(48z 2 + 144z + 113) + δ 2 (4z + 5)2 (4z + 11)2 .
√√48z 2 +144z+113−4 2 (2z+3)2 (4z+5)(4z+11)
Solving L2 ≥ 0 for δ, we obtain δ ≤ δx ≡; δx is increasing(4z+5)(4z+11)√ )(for z, limz→∞ δx = 3 − 2 2 ≃ 0.171573, and δx = 0 for z = z1 ≃ 5.90928. Q.E.D.
Proof of Proposition 1. (i) Diﬀerentiating (2) and (3) with respect to z yields
2
2
2
2
∂t∗
= 16(a−c)[9(23δ −18δ+55)+16(7δE 2−2δ+15)z +8(37δ −22δ+85)z] > 0,∂z
2
2 +2δ+5)z 2 +8(25δ 2 +2δ+25)z]∂qii∗= 16(a−c)[125δ −38δ+125+16(5δ
> 0,
2
E
∂z
∗
∂qij
2
2 −2δ+5)z(3+z)]= − 32(a−c)[41δ −62δ+185+16(δ
< 0.
2
E
∂z

(ii) Diﬀerentiating (1) with respect to z yields ∂x∗i /∂z = [128(a − c)(2z + 3)/E 2 ][δ(8z + 7) − 5],which implies part (ii). Q.E.D.
Proof of Proposition 2. Since CSi∗ = (Q∗i )2 /2, sign{∂CSi∗ /∂z} = sign{∂Q∗i /∂z}. The diﬀerentiation of total output yields ∂Q∗i /∂z = 16(a−c)[(48z 2 + 104z + 43)(δ 2 + 2δ) − 5(4z + 7)2 ]. Thus,E2√√
4 2 (2z+3)2 (48z 2 +104z+43)(> 0); δcs is decreasing for z and∂Q∗i /∂z ≥ 0 for δ ≥ δcs ≡ −1 +48z 2 +104z+43
δcs = 1 for z =(√)/
30 − 1 4 ≃ 1.11931. Q.E.D.

Proof of Lemma 3. I. From Π∗i , we obtain ∂Π∗i /∂z = [512(a − c)2 (2z + 3)2 /E 3 ][3(47δ 2 − 56δ +25) + 16(7δ 2 − 4δ + 5)z 2 + 16(17δ 2 − 13δ + 10)z] > 0 and

29696z 5 +221952z 4 +678272z 3 +1054112z 2 +829172z+263171


2(a − c)2 (4z + 5) ∂Π∗i
 + δ 3 (4z+5)2 (4z+11)3 −3δ 2 (4z+5)(4z+11)2 (48z 2 +144z+113)  > 0.
=
3


∂δ
E
− δ(4z + 7)(4z + 11)(832z 3 + 3024z 2 + 3756z + 1563)

20

II. (i) From πk∗ and∑n∗∗
k=1 πk = nπk , we obtain222
2 −2δ+15)z 2 +8(31δ 2 −28δ+85)z]∂πk∗= − 256(a−c) (2z+3) [3(55δ −58δ+175)+16(7δ
< 0,
2E3
n
∂n
2
2
2
2 −2δ+5)z 2 +16(2δ 2 −7δ+15)z]∂πk∗= − 512(a−c) (2z+3) [3(9δ −22δ+65)+16(δ
< 0.
2E3
n
∂λ
∑
( ∗)
∂ ( nk=1 πk∗ )
2 ∂πk
=n
< 0.
∂z
∂λ

(ii) From πk∗ ,]2 (2z+3)3 [∂πk∗= 1024(a−c)
16z 2 + 40z + 29 − (4z + 5)(4z + 11)δ ,3nE∂δwhich implies part (ii). Q.E.D.
Proof of Proposition 3.

(i) Diﬀerentiating (4) with respect to z yields ∂SW ∗i /∂z = [128(a −c)2 (2z + 3)/E 3 ]N , where N ≡ (1088z 3 + 4272z 2 + 5292z + 1993)δ 2 − 2(4z + 7)(16z 2 + 88z +101)δ + 5(64z 3 + 112z 2 − 84z − 163). N > 0 if z > 1.37301, therefore ∂SWi∗ /∂z > 0 ∀z ≥ 3/2.
(ii) Diﬀerentiating (5) with respect to z yields∂W W ∗
256(a−c)2 (2z+3)=∂zE3Thus, ∂W W ∗ /∂z ≥ 0 iﬀ δ ≥ δ[](1024z 3 +4048z 2 +4992z+1831)δ 2.

−2(144z 2 +552z+509)δ − 5(176z 2 +528z+397)√

509+552z+144z 2 +4 (3+2z)2 (27039+70232z+55984z 2 +14080z 3 ). Sinceww ≡1831+4992z+4048z 2 +1024z 3√305871
δww is monotonically decreasing for z, limz→∞ δww = 0, and δww z=3/2 = 1661+24
≃
21883
(
]
√
305871∀z ≥ 3/2. Q.E.D.
0.682463, δww ∈ 0, 1661+24
21883

Proof of Proposition 4. Diﬀerentiating (7) with respect to z yields∂Qd∗9(a−c)
i
=
∂z
K2

[

z(92z 3 +200z 2 +147z+36)δ 2 +z(7z+4)(2z+3)2 δ−(4z + 3)(28z 3 + 52z 2 + 36z + 9)ddThus, ∂Qd∗i /∂z ≥ 0 if δ ≥ δcs , where δcs ≡√√3z(2z+1)2 M1 −z(7z+4)(2z+3)22z(92z 3 +200z 2 +147z+36)].

and M1 ≡ 3500z 5 +d ,13388z 4 + 21243z 3 + 17496z 2 + 7452z + 1296. From the equation of δcs(√ √)2
3 z(2z + 1)2 M1 − [z(2z + 3)2 (7z + 4)]2
= 4z(4z + 3)(28z 3 + 52z 2 + 36z + 9)(92z 3 + 200z 2 + 147z + 36) > 0,

21

√d > 0. We find that δ d → 5 105 −7 ≃ 0.961625 as z → ∞, and δ d − 1 ≤ 0 for z ≥ z ≃so δcs2cs
cs
46

2.58114. Q.E.D.
Proof of Proposition 5. Diﬀerentiating SWid∗ with respect to z yields∂SWid∗
27(a − c)2 (2z + 1)
=
3
∂z

 ( K
)
1656z 6 +8292z 5 +15150z 4 +13471z 3 +5985z 2 +1161z+54 δ 2
 () 
6 +5492z 5 +10586z 4 +9687z 3 +4833z 2 +1323z+162 δ  .
×
−
840z


−384z 6 −632z 5 −776z 4 −1498z 3 −1638z 2 −783z−135
d , where
Solving ∂SWid∗ /∂z ≥ 0 for δ, we obtain δ ≥ δsw

d
δsw
≡

√
M2 + (2z + 1)(2z + 3) M3,
2(1656z 6 +8292z 5 +15150z 4 +13471z 3 +5985z 2 +1161z+54)

M2 ≡ 840z 6 + 5492z 5 + 10586z 4 + 9687z 3 + 4833z 2 + 1323z + 162, and M3 ≡ 203076z 8 +822036z 7 + 1677397z 6 + 2360146z 5 + 2340511z 4 + 1501530z 3 + 560061z 2 + 103248z + 6156. We√

d → 35+ 5641 ≃ 0.797874 as z → ∞, and δ d − 1 ≤ 0 for z ≥ ẑ ≃ 0.569. Hence,find that δsw
sw
138
d ∈ (0, 1) ∀z ≥ 3/2. Q.E.D.
δsw

B2. The four eﬀects in (6)(i) Domestic supply eﬀect (term (i)) is2∂qii[pi −(c−(1+δ)xi )]= 128(a−c) (2z+3)(4z+5)
E3
∂z

[

125δ 2 −38δ+125+16(5δ 2 +2δ+5)z 2]> 0.

+ 8(25δ 2 +2δ+25)z(ii) Investment eﬀect (term (ii)) is[(1+δ)Qi −2xi ]2∂xi= 256(a−c) (2z+3)[δ(8z+7)−5][139δ−29+16(3δ−1)z 2 +8(21δ−5)z] ⋛ 0.
E3∂z(iii) Export eﬀect (term (iii)) is
2 (2z+3)∂qij= − 512(a−c)[pj −(c−(1+δ)xi )−t]
3
E
∂z

[

41δ 2 −62δ+185 + 16(δ 2 −2δ+5)z 2]+ 48(δ 2 − 2δ + 5)z(iv) Transport price eﬀect (term (iv)) is
2 (2z+3)∂t−qij= − 256(a−c)
3
E
∂z

[

9(23δ 2 − 18δ + 55) + 16(7δ 2 − 2δ + 15)z 2+ 8(37δ 2 − 22δ + 85)z
22

]
< 0.

< 0.

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26

1.0
0.8

0.025

∆cs

¶ Qi*
>0
¶z

∆

0.6

*
¶ qji
¶z

0.020

0.4

0.015

¶ Qi*
<0
¶z

0.2

-

*
¶ qii
¶z

0.010

0.0
2

3

4

5

6

0

7

z

1
4

1
2

3
4

1

∗ /∂z)
Panel (b): ∂qii∗ /∂z and −(∂qji(z = 3; a − c = 1).

Panel (a): The area “∂Q∗i /∂z > 0.”

Figure 1: Illustration of Proposition 2.

0.015

(i)

0.010
0.005

(ii)
2

2.5

z

3

-0.005

(iii)

(iv)

-0.010Figure 2: Graph of “(each term)/(a − c)2 .” (δ = 1/4)

27

∆

∆
1.0
0.9
0.8
0.7

3
2

5

8

11

z

d (black curve) and δ (gray curve).
Panel (a): Two thresholds: δcs
cs

0.014
0.012
0.010
0.008
0.006
0.004
0.002
1
4

1
2

3
4

1

∆

∗ /∂z) (dashed gray curve),
Panel (b): ∂qii∗ /∂z (gray curve), −(∂qjid∗ /∂z) (dashed black curve).
∂qiid∗ /∂z (black curve), and −(∂qji
Note: These four curves are illustrated where z = 5; a − c = 1.

Figure 3: Comparison of two cases: price-taker and duopsony.

28

Supplementary Material (Not for publication)
This supplement provides two additional results (Lemmas 4 and 5) in section 4.
Lemma 4 t∗ > td∗ ∀z ≥ 3/2.
Proof. The diﬀerence between t∗ and td∗ is equal tot∗ − td∗ =2(a − c)(2z + 3)
[φ(δ)] ,
EK

where
(
)
φ(δ) ≡ 180 + 627z + 838z 2 + 824z 3 + 384z 4 − 2 36 + 111z + 152z 2 + 148z 3 + 48z 4 δ()+ 72 + 231z + 154z 2 − 64z 3 − 96z 4 δ 2 .
Thus, sign{t∗ − td∗ } = sign{φ(δ)}.
Let A ≡ 72 + 231z + 154z 2 − 64z 3 − 96z 4 in φ(δ). Then, from numerical calculation,
A > (=, <) 0 for z < (=, >) zA ≃ 1.55637. To show that φ(δ) > 0 ∀z ≥ 3/2, we need to consider
the sign of φ(δ) in the following three cases: (i) A = 0, (ii) A > 0, and (iii) A < 0.
Case (i) A = 0. When z = zA , φ(δ)|z=zA is monotonically decreasing with respect to δ.
Since φ(δ = 1)|z=zA = 5712.19 > 0, t∗ > td∗ .
Case (ii) A > 0. In this case, φ(δ) is downward convex. Solving φ(δ) = 0 for δ, we obtain√ √(36 + 111z + 152z 2 + 148z 3 + 48z 4 ) ± 2 3 (2z + 1)Γ
δ ,δ =
72 + 231z + 154z 2 − 64z 3 − 96z 4+−

and Γ ≡ 1632z 7 + 4096z 6 + 2558z 5 − 3201z 4 − 7992z 3 − 8235z 2 − 4617z − 972. From numericalcalculation, Γ ≤ 0 for z ≤ ž ≃ 1.37274, so Γ > 0 for all z ≥ 3/2.
δ + and δ − yield√ √{}2 (72z 4 + 106z 3 − z 2 − 60z − 18) + 3 (2z + 1)Γ
δ −1=
,
A
√
√
{
}
2 (72z 4 + 106z 3 − z 2 − 60z − 18) − 3 (2z + 1)Γ
δ− − 1 =.
A+

72z 4 + 106z 3 − z 2 − 60z − 18 > 0 for all z ≥ 3/2, so δ + − 1 > 0. Further,(72z 4 + 106z 3 − z 2 − 60z − 18

)2

−

(√ √
)2
3 (2z + 1)Γ
()= 45 + 159z + 172z 2 + 116z 3 + 48z 4 A > 0. (S1)Thus, (S1) yields δ − − 1 > 0, which implies that t∗ > td∗ for all 3/2 ≤ z < zA .
Case (iii) A < 0. Since its numerator is positive, δ + < 0. By contrast, from the numeratorof δ − , we obtain(48z 4 + 148z 3 + 152z 2 + 111z + 36

)2

−

(√ √
)2
3 (2z + 1)Γ
()= 384z 4 + 824z 3 + 838z 2 + 627z + 180 A < 0.

Hence, δ − > 0. On the one hand, from (S1), the numerator of δ − − 1 is negative, and its
denominator, A, is also negative, so δ − − 1 > 0. Therefore, t∗ > td∗ for all z > zA . Q.E.D.

Lemma 5 ∂Πd∗i /∂z > 0 ∀z ≥ 3/2.
Proof. Diﬀerentiating Πd∗i with respect to z, we obtain∂Πd∗27(a − c)2 (2z + 1)2i=[ξ(δ)] ,
∂z
K3
where
ξ(δ) ≡ 144z 5 + 1216z 4 + 2138z 3 + 1440z 2 + 378z + 27
− 3(2z + 1)(3z 2 + 13z + 6)(28z 2 + 30z + 9)δ
+ (552z 5 + 2672z 4 + 4222z 3 + 2961z 2 + 837z + 54)δ 2 .
The ξ(δ) is downward convex. The discriminant of “ξ(δ) = 0” is (2z +3)2 (−15984z 8 −258096z 7 −785156z 6 − 990908z 5 − 561419z 4 − 82110z 3 + 50841z 2 + 21600z + 2268) and it has a negativevalue for any z > 0.332382, so ξ(δ) = 0 does not have real roots and ξ(δ) > 0 ∀z ≥ 3/2. Thisimplies ∂Πd∗i /∂z > 0. Q.E.D.