International R&D rivalry with a shipping firm
Kazuhiro Takauchi†This version: January 5, 2013

Abstract

The purpose of this paper is to consider the role of a monopolistic shipping ﬁrm in international research and development (R&D) rivalry. In a twocountry, two-ﬁrm duopoly with one-way trade, the exporter competes withthe local ﬁrm in the local market but must pay a per-unit shipping fee to theshipper in order to carry its product. We show that the investment level ofthe exporter may decrease as the eﬃciency of R&D improves. To reduce theshipping fee, the exporter commits itself to a smaller investment level but it
raises the rival’s investment. For this reason, the exporter’s investment may
decrease as the eﬃciency of R&D improves. We further investigate a case of
two-way trade comprising two symmetric countries. When a single R&D sector
undertakes investment for domestic supply and export, producers’ investment
increase as the eﬃciency of R&D improves. We show that, in this case, theproducers’ proﬁt may decrease as the eﬃciency of R&D improves.
Keywords: Shipping ﬁrm, Shipping fee, Eﬃciency of R&D, Exports
JEL Codes: F12, L13, L91†

Corresponding author: Faculty of Economics, Management and Information Science,

Onomichi City University, 1600-2 Hisayamada-cho, Onomichi-city, Hiroshima 722-8506, Japan.
E-mail address: kazh.takauch@gmail.com
Tel.: +81-848-22-8311; Fax: +81-848-22-5460

1

Introduction

In international competition among ﬁrms, what factors aﬀect exports? At present, it isdiﬃcult to freely raise the rate of import tariﬀs for many countries, and tariﬀ rates are
low enough.1 Historically, the tariﬀ has been considered an important trade cost, but
nowadays, this is not necessarily true: a main that prevents export activity is not importtariﬀ but transport price.2 As shown by Hummels (2007) and Hummels et al. (2009),
maritime transport price has a particularly important role in exports. First, as pointed
out by Hummels (2007), the air transport price smoothly decreases, but the ocean freightrate does not decrease over time. Second, as empirically shown by Hummels et al. (2009),
the ocean freight rate is a mark-up price, and as such, the shipping ﬁrm has a certain
monopoly power.
The other factor that possibly aﬀects ﬁrm export activity is its productivity. In general, the productivity of a ﬁrm can be improved by its cost-reducing innovation. To invest
in cost-reducing research and development (R&D), ﬁrms utilize a smaller level of the production cost.3 Because a smaller production cost directly corresponds to a larger volume
of output, ﬁrms can export larger volumes than earlier. For example, as empirically shownby Yasar et al. (2006), ﬁrm-level productivity enhancement results in higher export performance.4 Moreover, less expenditure into R&D reduces production cost if the eﬃciency
of R&D improves. Thus, the eﬃciency of investment possibly aﬀects ﬁrm export.
The purpose of this paper is to consider the relationship among endogenously decided
shipping fee, cost-reducing innovation, and ﬁrm export. To examine the strategic interaction among them, we introduce a monopoly shipper in an international cost-reducing R&D
rivalry model. There are two producers, an exporter and a local ﬁrm, and a non-producer,

1
Anderson and Wincoop (2004) indicate that the arithmetic average rate of tariﬀ is less than 5% forrich countries.
2
Clark et al. (2004) point out that transport prices are a greater barrier to U.S. markets than importtariﬀs for most Latin American countries.
3
There are many studies on international R&D rivalry. For example, see Spencer and Brander (1983),
Leahy and Neary (1996, 1999), Qiu and Tao (1998), Dewit and Leahy (2004), Kujal and Ruiz (2007, 2009),
Haaland and Kind (2008), and Takauchi (2011). They focus on the eﬀects of R&D subsidy policy, thespillover eﬀects of investment, and the interactions between other trade policies and ﬁrms’ R&D incentives.
4
They examine the relationship between export and productivity for Turkish manufacturing industries.

1

a shipping ﬁrm. The exporter and the local ﬁrm compete à la Cournot in the market ofthe local country and engage in cost-reducing R&D activity, whereas the monopoly shipper carries the product produced by the exporter to the market of the local country. The
timing of the game is as follows. In the ﬁrst stage, each producer invests in cost-reducing
R&D, respectively. In the second stage, the shipper decides the shipping fee. In the ﬁnal
stage, each producer decides the quantity of its product.
We show that if R&D is highly eﬃcient, the exporter’s investment decreases as theeﬃciency of R&D improves. The key to this result is that the shipping fee decreases
as the exporter’s investment decreases. For this reason, to reduce the fee, the exporter
commits to a smaller level of investment. Because this commitment also increases therival’s investment level, the exporter’s investment decreases when the rival has larger
incentives to invest.
We further extend the above one-way trade to a two-way (or intra-industry) tradecomprising two symmetric countries. In contrast to the above one-way trade, the producer
located in each country has two diﬀerent plants: one for domestic supply and the otherfor export. When the producer located in each country exports to the other country, it
must pay a shipping fee to the monopoly shipper. We show that if the result of R&D canbe commonly used across diﬀerent plants (i.e., a single R&D sector can implement costreducing innovation across the two plants), the producers’ proﬁt decreases as the eﬃciency
of R&D goes up over a critical level. In this case, producers do not commit to a smaller levelof investment in order to reduce the shipping fee. Thus, equilibrium investments increaseas the eﬃciency of R&D improves. This yields that an increase in the equilibrium total
outputs is smaller than that in investments. Since the diﬀerence between them becomeslarger when R&D is highly eﬃcient, the producers’ proﬁt decreases when the eﬃciency goesup over a critical revel. In a usual cost-reducing R&D rivalry, the investments, outputs,
and proﬁt increase as the eﬃciency of R&D improves. However, our results imply that
producers’ technology improvement does not always enhance their activities; in particular,
an improvement in production eﬃciency may reduce the producers’ proﬁt. This can beseen in the strategic interaction between the producers’ technology improvement and the2

endogenous freight rate, and thus, we can say that our results provide a new insight inthe study of transportation and international trade.
A number of works focus on international transportation and freight rate (Francois
and Wooton, 2001; Behrens and Picard, 2011; Kleinert and Spies, 2011).5 Although thesestudies are seminal works that focus on the market power of the transport industry, they
do not examine the relationship between the producer’s process innovation and freightrates.
This paper is related to some studies on upstream-downstream oligopoly with R&D(Banerjee and Lin, 2003; Ishii, 2004; Mukherjee and Ray, 2007; Matsuhima and Mizuno,2012). They consider the cost-reducing innovation of input suppliers or ﬁnal-good manufactures in a vertically related market. In these vertical structures of production, to
produce a ﬁnal-good, the downstream agent always purchases the input from the upstream supplier. However, in our international transporting structure, the producer paysa per-unit shipping fee when exporting its product to the designation country. Thus, thesemodels of upstream-downstream oligopoly do not cover international transportation.
Some works on unionized oligopoly with R&D are also related to this paper (Calabuigand Gonzalez-Maestre, 2002; Haucap and Wey, 2004; Mukherjee et al., 2007; Manasakis
and Petrakis, 2009; Mukherjee and Pennings, 2011). In our model, the shipper sets thefreight rate in a manner similar to that employed in the above labor union model: theshipper decides the freight rate (the per-unit fee on shipment) and the labor union decides
the wedge rate (the per-unit fee on the usage of labor). However, our model is cruciallydiﬀerent from the above labor union model. First, in the above studies, it is assumed thateach ﬁrm has a union or that all ﬁrms have a common union. In our model, the only
exporter (or the plant exporting) must use the shipper and pays the freight rate. This
point is fundamentally diﬀerent from these studies.6 Second, they focus on the role of5

There is an another study which focuses on the international transportation. Matsushima and Takauchi(2012) focus on the role of ports and international shipment in a two-way oligopoly. They examine thatthe choice of ownership structure in an international port (privatization or nationalization) possibly works
as a strategic trade policy, which controls trade volumes.
6
Although Calabuig and Gonzalez-Maestre (2002) and Haucap and Wey (2004) consider a unionizedindustry, they do not examine a standard cost-reducing R&D rivalry. In particular, Haucap and Wey (2004)
assume a patent tournament of two ﬁrms: they have a chance to get exclusive rights to an innovation,
which can reduce the labor requirement. In contrast, Mukherjee et al. (2007) and Mukherjee and Pennings

3spillover eﬀects of investment, but tend to neglect the roll of the eﬃciency of R&D.7
In the next section, we build a simple one-way trade model. Section 3 derives theequilibrium outcomes and examines ﬁrms’ investment and output strategies and section 4
discusses the case of Brander and Krugman (1983)-type two-way trade. Section 5 oﬀers aconclusion.

2

Model

We consider a one-way trade model with a shipping ﬁrm. There are two countries, local
and foreign. The local country has a product market, but the foreign one does not. We callthe foreign country an exporting country. Two producers (an exporter and a local ﬁrm)
and a shipper exist in the world, and the exporter (local ﬁrm) is in the exporting (local)
country. The exporter and the local ﬁrm compete à la Cournot in the local country’s
market. We assume that the local ﬁrm supplies its product without a shipping fee. In
contrast, since the exporter is located overseas, it must use the shipper to carry its productto the local market across the ocean. To transport the product, the exporter must pay
the per-unit shipping fee, f , to the shipper.8
Before market competition, the exporter and the local ﬁrm engage in cost-reducing
R&D competition without spillover. The unit production cost is c (> 0), and is the samefor both the exporter and the local ﬁrm.9 To reduce the initial unit cost, all producers
make cost-reducing investments. After investment, the unit cost becomes c − xi , wherexi is the investment level of producer i. Variables associated with the two producers are
subscripted by i = E, L (exporter and local ﬁrm). We assume that the R&D cost functionis (γ/2)(xi )2 , where γ is the eﬃciency of R&D and a positive constant.10 That is, a smaller(2011) mainly focus on the technology licensing of downstream manufactures.
7
For example, Manasakis and Petrakis (2004) consider the relationship between the spillover eﬀects ofinvestment and the formation of research joint ventures in duopoly ﬁrms.
8
As empirically demonstrated by Hummels et al. (2004), transport prices are not ad valorem (i.e.,
iceberg)-type but per-unit type.
9
Because our focus is not on cost asymmetry, we assume that the two producers have the same production technology.
10This type of R&D cost function is popular. For example, see Leahy and Neary (1996, 1999), Qiu
and Tao (1998), Dewit and Leahy (2004), Motta (2004), Kujal and Ruiz (2009), Takauchi (2011), andMatsushima and Mizuno (2012).

4

γ corresponds to a higher eﬃciency in R&D.
The inverse market demand function in the local country is given by
p = a − qE − qL ,where p is the product price, qE is the export quantity of the exporter, qL is the outputof the local ﬁrm, a is a positive constant, and a > c. Thus, the proﬁt of producer i, Πi , isgiven byΠE ≡ [a − (qE + qL ) − (c − xE ) − f ] qE −
ΠL ≡ [a − (qE + qL ) − (c − xL )] qL −γ(xE )2 ,2γ(xL )2 .
2

(1)
(2)

The shipper makes a take-it-or-leave-it oﬀer to the exporter and decides the per-unit
shipping fee, f . The proﬁt of the shipper, πS , is given asπS ≡ (f − f0 ) qE ,(3)

where f0 is the minimum price of shipment. For simplicity, we normalize f0 to zero.11
To examine the strategic interaction between the shipper and producers, we consider
the following three-stage game. At stage 1, the exporter and local ﬁrm independently
and simultaneously choose a volume of R&D investment. At stage 2, the shipper sets theshipping fee. At stage 3, the exporter and local ﬁrm compete à la Cournot in the local
market.
This timing of the game reﬂects the planning time relevant to players’ decisions. In
general, the decision on investment has a much longer time while the shipping fee isnegotiated for a shorter time. Since quantities in the product market can be adjusted forthe shortest time among all decisions, the decision on production quantities is done in theﬁnal stage of the game.
We use the subgame perfect Nash equilibrium as the equilibrium concept. The game11

This speciﬁcation does not alter our results.

5

is solved by backward induction.

3

Equilibrium outcomes in one-way trade

In this section, we derive the equilibrium outcomes and examine the strategies of the
producers and the shipper.
Third stage. Two producers compete à la Cournot in the market of the local country.
Using (1) and (2), the FOC for the proﬁt maximization of each producer isExporter:

0 = a − c − f − 2qE − qL + xE ,Local ﬁrm:

0 = a − c − qE − 2qL + xL .

The above equations yield the following outputs in the third stage:qE (xE , xL , f ) =a − c − 2f + 2xE − xLa − c + f − xE + 2xL; qL (xE , xL , f ) =
.
3
3

(4)

Second stage. In the second stage, the shipper chooses a level of the shipping fee, f .
From (3) and (4), the shipper maximizes f qE (xE , xL , f ) for the shipping fee. The FOC is21
0 = − f + (a − c − 2f + 2xE − xL ).
3
3Solving the FOC for f , we obtain the shipping fee in the second stage:f (xE , xL ) =a − c + 2xE − xL
.
4

(5)

Thus, the shipping fee rises (falls) as the volume of R&D investment by the exporter
increases (decreases).
In the second stage, from (4) and (5), the outputs of the producers areqE (xE , xL ) =5(a − c) − 2xE + 7xL
a − c + 2xE − xL
; qL (xE , xL ) =
.
6
12

6

(6)

First stage. Here, let the asterisks denote the subgame perfect Nash equilibrium of thegame. In the ﬁrst stage, producers choose their respective volumes of R&D investment.
Using (1), (2), (5), and (6), the FOCs in this R&D stage are
Exporter:

0 = a − c + (2 − 9γ)xE − xL ,Local ﬁrm:

0 = 35(a − c) − 14xE + (49 − 72γ)xL .

The above equations yield the equilibrium levels of R&D investments, x∗E and x∗L (equation(8), below). The equilibrium outputs, investments, and proﬁt of the producers are6(a − c)γ(15γ − 4)
6(a − c)γ(6γ − 7)
; qL∗ =,
228 − 195γ + 216γ
28 − 195γ + 216γ 24(a − c)(6γ − 7)
7(a − c)(15γ − 4)x∗E =; x∗L =,
28 − 195γ + 216γ 2
28 − 195γ + 216γ 24(a − c)2 (6γ − 7)2 γ(9γ − 2)(a − c)2 (15γ − 4)2 γ(72γ − 49)
∗
ΠE∗ =
;
Π
=
.
L
(28 − 195γ + 216γ 2 )22(28 − 195γ + 216γ 2 )2
∗
qE
=

(7)
(8)
(9)

The equilibrium shipping fee and the proﬁt of the shipper are9(a − c)γ(6γ − 7)54(a − c)2 γ 2 (6γ − 7)2∗f =.
; πS =
28 − 195γ + 216γ 2(28 − 195γ + 216γ 2 )2∗(10)

We make the following two assumptions.
Assumption 1. The eﬃciency of R&D is not too high, i.e., γ > γ ≡ 7/6.
To ensure positive quantities for all outcomes in the game, we require this assumption.
Note that this is a stronger condition on γ than the condition required to satisfy the SOCs
in both the R&D and fee-setting stages.12 Thus, all other conditions are satisﬁed as longas Assumption 1 holds.
Assumption 2. The unit production cost, c, satisﬁes c/(a − c) > x∗L /(a − c).
To ensure that the ex-post unit production cost has a positive value (i.e., c − x∗i > 0),
The SOC for proﬁt maximization in the R&D stage is γ > 2/9 ≃ 0.22222 for the exporter andγ > 49/72 ≃ 0.680556 for the local ﬁrm.
127

it suﬃces to consider Assumption 2. This is because among all of R&D investments, thebiggest R&D investment is x∗L and it is monotonically decreasing with γ.13
The above (7)–(9) yield the following proposition.
Proposition 1. (i) When R&D is highly eﬃcient, the volume of R&D investment of
the exporter decreases as the eﬃciency of R&D improves; otherwise, it increases as the√eﬃciency of R&D improves, i.e., ∂x∗E /∂γ > 0 if γ < γb ≡ (1/12)(3 7 + 14) ≃ 1.8281 but∂x∗E /∂γ ≤ 0 if γ ≥ γb. However, the volume of R&D investment of the local ﬁrm increases
as the eﬃciency of R&D improves. (ii) The output and proﬁt of the exporter decrease, but
those of the local ﬁrm increase as the eﬃciency of R&D improves.
Proof. See Appendix A.
Figure 1 illustrates the shapes of the equilibrium investments of the exporter and the
local ﬁrm for γ (equation (8)).
[Insert Figure 1 here]
In a usual cost-reducing R&D competition, the volume of investment increases as theeﬃciency of R&D improves.14 Thus, one might think that the result shown in Proposition
1 is in sharp contrast to the intuition. When R&D is highly eﬃcient, why does theexporter reduce the volume of investment as the eﬃciency of R&D improves? The logic
behind this result depends on two factors. The ﬁrst is that the economy of scale works in
R&D investment; the eﬀect is stronger if R&D is highly eﬃcient. The second is that R&D
investment works as a commitment device for the shipping fee; to commit to a smaller
level of investment reduces the shipping fee, which decreases as the exporter’s investment
level decreases.
In the R&D stage, each producer will invest heavily if it believes it can sell heavily.
Furthermore, since a smaller γ corresponds to a higher R&D eﬃciency, investment incentives are larger. The exporter must pay the shipping fee, and thus, it has a smaller
13
14

For example, when we set a − c = 1, c ≥ 1 suﬃces Assumption 2 for any γ > γ.
For example, see Motta (2004, Ch. 2).

8market share than the local ﬁrm. That is, since the local ﬁrm has an advantage in market
competition, the local ﬁrm invests actively if the eﬃciency of R&D is high (i.e., γ is low).
The second factor promotes the eﬀects of the scale economy. Because the investment
level is decided in the ﬁrst stage of the game, the exporter can control the shipping fee.
From (5), a reduction in xE directly reduces the shipping fee and increases the local ﬁrm’s
investment level (i.e., strategic substitutes in the R&D stage). To reduce the shipping fee,the exporter always commits to a smaller level of investment; this behavior induces a larger
level of investment by the local ﬁrm. When the eﬃciency of R&D is high, the eﬀects ofa smaller investment considerably increase the local ﬁrm’s investment and further reducethe investment level of the exporter. Therefore, if γ becomes smaller than the criticallevel, γb, then the investment level of the exporter falls.
From (10), we establish the following proposition.
Proposition 2. (i) The shipping fee rises and becomes (1/4)(a − c) as the eﬃciency of
R&D becomes worse. (ii) The proﬁt of the shipper increases as the eﬃciency of R&D
becomes worse.
Proof. (i) Diﬀerentiating f ∗ with respect to γ, we obtain∂f ∗18(a − c)(−98 + 168γ + 171γ 2 )=.
∂γ(28 − 195γ + 216γ 2 )2√
From the numerator of the above equation, −98+168γ+171γ 2 ≥ 0 for all γ ≥ (7/57)(3 6−4) ≃ 0.411. Hence, ∂f ∗ /∂γ > 0. Rearranging f ∗ , we have limγ→∞ f ∗ = (1/4)(a − c).
(ii) Diﬀerentiating πS∗ with respect to γ, we obtain∂πS∗
216(a − c)2 γ(6γ − 7)(−98 + 168γ + 171γ 2 )
=
> 0.
∂γ
(28 − 195γ + 216γ 2 )3These results imply Proposition 2. Q.E.D.
Figure 2 illustrates the equilibrium shipping fee and the proﬁt of the shipper for γ(equation (10)).

9

[Insert Figure 2 here]

The reason is explained by a simple mechanism. In Proposition 1, we ﬁnd that theoutput of the exporter decreases as the eﬃciency of R&D improves. This reduction inexport volume corresponds to the shipping fee. This is because the shipper and theexporter have a certain cooperative relationship. Since a shipper’s proﬁt directly depends
on the output of the exporter, to ensure positive production for the exporter, the shipper
reduces its fee when the exporter is extremely weak. Hence, the proﬁt of the shipper also
goes down as the output of the exporter goes down.
In a cost-reducing R&D competition, investment is not performed if the cost of R&D
increases indeﬁnitely (i.e., xi → 0 as γ → ∞). That the shipping fee has an upper limit
corresponds to this fact.
In addition, we immediately ﬁnd the following result.
∗ +q ∗ , increase as the eﬃciency
Lemma 1. Total investment, x∗E +x∗L , and total output, qE
Lof R&D improves.
Proof. From (8), diﬀerentiating x∗E + x∗L with respect to γ, we obtain∂(x∗E + x∗L )36(a − c)(203 − 672γ + 774γ 2 )
=−
< 0.
∂γ
(28 − 195γ + 216γ 2 )2
∗ + q ∗ with respect to γ, we obtain
From (7), diﬀerentiating qEL∗ + q∗ )∂(qE
6(a − c)(308 − 1176γ + 1719γ 2 )
L
=−
< 0.
∂γ
(28 − 195γ + 216γ 2 )2Therefore, Lemma 1 holds. Q.E.D.
Lemma 1 implies that the local ﬁrm plays a major role but the exporter plays a minor
role in view of the local country’s domestic welfare. Because the exporter must pay theshipping fee and the market share of the ﬁrm is considerably smaller than that of thelocal ﬁrm, the eﬀects of a change in γ on the local ﬁrm undoubtedly dominate that on theexporter. For this reason, the consumers’ surplus in the local country strictly increases10

as the eﬃciency of R&D improves. Further, the domestic welfare of the local country iscomprised by the consumers’ surplus and the local ﬁrm’s proﬁt, and as such, the welfare
also increases as the eﬃciency of R&D improves.15

4

Two-way tradeThis section refers to Brander and Krugman (1983)-type two-way (or intra-industry) trade.
In the previous section, we have been obtained interesting results (propositions 1 and 2).
However, do these results also hold in the case of two-way trade?
As it turns out, our results of one-way trade do not depend on the trade structure,whether one-way or two-way. The key factor that controls our results is whether theinvestment in R&D works as a commitment device for the shipper (decision of fee-setting)or not. In two-way trade with process innovation, each producer located in each country
has two diﬀerent plants: one for domestic supply and the other for export. If each plantinvests, the investment for exporting can work as the commitment device for the shipper.
When we consider the technical training and skills of the employees of each plant, theenhancement in these employee attributes (the result of cost-reducing R&D) is exclusively
used in that plant. In addition, each plant is operated by a speciﬁc technique unique
to it is the same. In these cases, since it is diﬃcult to use common knowledge across
two diﬀerent plants, each plant invests to reduce its own production cost. In contrast, if
the result of R&D can be commonly used across both plants (i.e., a single R&D sector
implements cost-reducing innovation across two plants), the investments do not work as acommitment device for the shipper.16
Hence, we consider the following two cases related to the result of R&D: one is that theresult of R&D cannot be commonly used across diﬀerent plants (Case 1), and the other isthat it can be commonly used (Case 2).
First, let us consider the following situation: there are two symmetric countries with ahomogeneous product market, Home (H) and Foreign (F ). In each country, one producer
15

The eﬀects of γ on the welfare of the local country are derived in Takauchi (2012).
It is often assumed that a single R&D sector implements cost-reducing innovation across two plants.
For example, see Haaland and Kind (2008).
16

11

exists and each producer i (= H, F ) produces a product for domestic supply and to export
it to another market (the market in each country is segmented). To carry its product tothe destination country, each producer pays a per-unit shipping fee, g, to the monopoly
shipper: the shipper charges a common shipping-fee for both producers. We deﬁne thatthe subscript represents the ﬂow of the product: “HH” denotes Home-to-Home and “HF ”
denotes Home-to-Foreign; that is, qHH is the domestic supply of producer H and qHF isthe export by producer H from Home to Foreign. The shipper transports qHF units ofthe product from Home to Foreign and qF H units of the product from Foreign to Home.
The proﬁt of the shipper is π S ≡ (qHF + qF H )g.
The inverse market demand function in each country is pi = a − qii − qji , where i ̸= j
and i, j = H, F . The timing of the game is the same as in the previous section.
Case 1. The result of R&D cannot be commonly used across diﬀerent plants:
In this case, R&D is implemented in each plant. The proﬁt of producer i (= H, F ) is givenbyγγ
Πi ≡ [pi − (c − xii )]qii − (xii )2 + [pj − (c − xij ) − g]qij − (xij )2 ,
2
2
|
{z
} |
{z
}

(11)

ExportDomestic supply

where i ̸= j and i, j = H, F .
Adopting a similar method as in the previous section, from (11), we obtain the following
equilibrium values:175(a − c)(5γ − 2)2(a − c)(4γ − 5)
4(a − c)γ(5γ − 2)
; xD
; qiiD =,ij =2
2
10 − 51γ + 48γ
10 − 51γ + 48γ
10 − 51γ + 48γ 22(a − c)γ(4γ − 5)(a − c)2 γ(−200 + 988γ − 1649γ 2 + 928γ 3 )
D
D
;
Π
=
qij
=
,
i
10 − 51γ + 48γ 22(10 − 51γ + 48γ 2 )2

xD
ii =

gD =

3(a − c)γ(4γ − 5);
10 − 51γ + 48γ 2πDS =12(a − c)2 γ 2 (4γ − 5)2(10 − 51γ + 48γ 2 )2

,

(12)

where superscript “D” denotes the equilibrium values in Case 1.18 To ensure positive
quantities, here, we need γ > 5/4.
17
18

For details of the derivation, see Appendix B.
In Case 1, we assume that c/(a − c) > xD
ii /(a − c).

12D /∂γ > 0, ∂g D /∂γ > 0, and ∂π D /∂γ > 0.19 Moreover, theFrom (12), we ﬁnd that ∂qijS

diﬀerentiation of xD
ij yields∂xD
2(a − c)(215 − 480γ + 192γ 2 )ij=−
for i ̸= j.
∂γ(10 − 51γ + 48γ 2 )2√D
Therefore, ∂xD
ij /∂γ >(≤) 0 if γ <(≥) γ ≡ (1/24)( 255 + 30) ≃ 1.91536.
Case 2. The result of R&D can be commonly used across diﬀerent plants:
Since a single R&D sector implements cost-reducing investment in both plants, the proﬁtof producer i is given byΠi ≡ [pi − (c − xi )]qii + [pj − (c − xi ) − g]qij −γ(xi )2 ,2(13)

where xi denotes the volume of investment by producer i.
From (13), the equilibrium values become as60(a − c)γ24(a − c)γ113(a − c)
C
; qij
=
; xC
,
i =
144γ − 113
144γ − 113
144γ − 113(a − c)2 γ(8352γ − 12769)
ΠiC =,2(144γ − 113)2
qiiC =gC =1728(a − c)2 γ 236(a − c)γ
; πC
=
,
S
144γ − 113(144γ − 113)2(14)

where superscript “C” denotes the equilibrium values in Case 2.20 To ensure positive
quantities, here, we need γ > 12769/8352 ≃ 1.5289.
C /∂γ < 0, ∂xC /∂γ < 0, ∂q C /∂γ < 0, andFrom (14), we ﬁnd that ∂qiiC /∂γ < 0, ∂qiji
21 In contrast, the derivation of producer’s proﬁt Π C yields∂π C
iS /∂γ < 0.

113(a − c)2 (12769 − 432γ)
∂ΠiC
=
.
∂γ
2(144γ − 113)3
From the above equation, ∂ΠiC /∂γ ≥ (<) 0 if γ ≤ (>) γ C ≡ 12769/432 ≃ 29.5579.
19

For these results, see Appendix D.
In Case 2, we assume that c/(a − c) > xCi /(a − c).
21
See also Appendix D.

20

13

Summarizing the above arguments, we establish the following result in the case ofsymmetric two-way trade.
Proposition 3. (i) Suppose that the result of R&D is commonly used across diﬀerent
plants (Case 2). Then, if γ < γ C ≡ 12769/432, the proﬁt of the producers decreases as theeﬃciency of R&D improves. (ii) Suppose that the result of R&D is not commonly used√
across diﬀerent plants (Case 1). Then, if γ < γ D ≡ (1/24)( 255 + 30), the producers’
R&D investments for exporting decrease as the eﬃciency of R&D improves; the shipping
fee and each producer’s exports decrease as the eﬃciency of R&D improves.

[Insert Figure 3 here]

Why does the proﬁt of the producers decrease as the eﬃciency of R&D improves? The
reason is that investments increase faster than total outputs for an improvement in theeﬃciency of R&D. To see this, we derive the following relation:
∆C
xq ≡

C)
∂(qiiC + qij∂∆C∂xC
1952640(a − c)6780(a − c)
xq
i
=−
−
>
0;
< 0.
=
2
∂γ
∂γ
∂γ
(144γ − 113)(144γ − 113)3

The result is explained by the following logic. The producers do not decrease investment
in order to reduce the shipping fee, and thus, the investment increases as the eﬃciency of
R&D improves (∂xC
i /∂γ < 0). This is because, reducing investments implies that they are
crowded out in the investment stage (strategic substitution of R&D rivalry), so producers
tend to increase those investments in equilibrium. This corresponds to the shipping fee
increasing as the eﬃciency of R&D improves (∂g C /∂γ < 0).
The producers’ outputs increase from the improved eﬃciency of R&D (∂qiiC /∂γ < 0 andC /∂γ < 0). Since the producers engage in Cournot competition, rising (reducing) rival’s∂qij

cost increases (decreases) one’s output. Therefore, an increase in the rival’s investment
reduces one’s domestic supply and export. One’s domestic supply increases as γ decreases,
but a reduction in γ increases the rival’s investment too.22 An increase in domestic supply
The third stage equilibrium values are qii (xi , xj , g) = (1/3)(a − c + g + 2xi − xj ) and qij (xi , xj , g) =(1/3)(a − c − 2g + 2xi − xj ) for i ̸= j. Further, see Appendix C.
22

14

from an improvement in the eﬃciency of R&D is not so large. One’s export also increases
as γ decreases, but a reduction in γ increases both the rival’s investment and the shippingfee. For this reason, an increase in export from an improvement in the eﬃciency of R&D
is small. Furthermore, if γ is small enough, the cost of investments becomes smaller andtheir enhancement becomes stronger. Therefore, if the eﬃciency of R&D goes up over a
critical level, γ C ≡ 12769/432 ≃ 29.5579, the producers’ proﬁt decreases.

5

Conclusion

This paper builds a model of international R&D rivalry with a monopoly shipper. Although the shipping fee (i.e., maritime transport price) and the productivity of the exporters are important factors that possibly aﬀect export behavior, the interactions among
them have not received much attention. We show that the exporter’s investment level may
decrease as the eﬃciency of R&D improves. The key to this result is whether the R&D
investments work as a commitment device for the shipper or not. To reduce the shippingfee, the exporter commits to a smaller investment level. Because this commitment raises
the rival’s investment level, the exporter’s investment decreases. In two-way trade, we
also ﬁnd that if the result of R&D is commonly used across diﬀerent plants (i.e., a single
R&D sector implements cost-reducing innovation across two plants), the proﬁt of the producers may decrease as the eﬃciency of R&D improves. When R&D is highly eﬃcient,
the increase in outputs is smaller than that of investments due to the improvement in the
R&D eﬃciency. Then, a higher eﬃciency of R&D corresponds to a smaller proﬁt for the
producers.
In process innovation, the eﬃciency of investment plays an important role. If eﬃciency
improves, investment incentives and output increase. This undoubtedly increases the proﬁtof the producers. However, our ﬁndings highlight some paradoxical results, which occur
if R&D is highly eﬃcient. Since all these ﬁndings depend on the behavior of the shipper,
we believe that our model developed herein oﬀers a new insight into transportation andinternational trade.
In this study, we have focused on the relationship between the productivity of the pro15

ducers and the shipping fee by considering these producers’ investment activities. However,one might think that the process innovation is not always done only by producers; nonproducers may do it. The strategic interaction of R&D activity among producers andnon-producers surpasses the scope of our analysis, and as such, we have not considered
the investment activity of the shipper. This issue may be fruitful for future research.

Acknowledgments
I thank Noriaki Matsushima for his many constructive comments. I also thank Seiichi
Katayama, Eiichi Miyagawa, Makoto Tawada, and the seminar participants at the second
spring meeting of the Japan Society of International Economics (Nanzan University) for
their helpful comments. Of course, all errors are my own.

Appendix
A. Proof of Proposition 1 (i) Diﬀerentiating (8) with respect to γ, we obtain∂x∗E
36(a − c)(133 − 336γ + 144γ 2 )=−,∂γ(28 − 195γ + 216γ 2 )2∂x∗L504(a − c)(5 − 24γ + 45γ 2 )=−.
∂γ(28 − 195γ + 216γ 2 )2√
From the numerator of ∂x∗E /∂γ, 133−336γ +144γ 2 ≥ 0 for γ ≥ (1/12)(3 7+14) ≃ 1.8281√√
and γ ≤ (1/12)(−3 7 + 14) ≃ 0.505229. This yields ∂x∗E /∂γ ≥ 0 if γ ≤ (1/12)(3 7 + 14),√but ∂x∗E /∂γ < 0 if γ > (1/12)(3 7 + 14). From the numerator of ∂x∗L /∂γ, we ﬁnd that
5 − 24γ + 45γ 2 > 0 for all γ. Thus, ∂x∗L /∂γ < 0. (ii) The diﬀerentiation of (7) yields∗∂qE
12(a − c)(−98 + 168γ + 171γ 2 )=,∂γ(28 − 195γ + 216γ 2 )2∂qL∗
6(a − c)(112 − 840γ + 2061γ 2 )=−.
∂γ(28 − 195γ + 216γ 2 )2√∗ /∂γ, −98 + 168γ + 171γ 2 > 0 for γ > (7/57)(3 6 − 4) ≃From the numerator of ∂qE∗ /∂γ > 0. From the numerator of ∂q ∗ /∂γ, we ﬁnd that 112 − 840γ +0.411216. Thus, ∂qE
L2061γ 2 = 0 has no real root and a positive value. Thus, ∂qL∗ /∂γ < 0. The diﬀerentiation16

of (9) yields∂ΠE∗
8(a − c)2 (6γ − 7)(196 − 903γ − 342γ 2 + 4374γ 3 )=,∂γ(28 − 195γ + 216γ 2 )3∂ΠL∗(a − c)2 (15γ − 4)(−5488 + 39648γ − 137277γ 2 + 138024γ 3 )=−.
∂γ2(28 − 195γ + 216γ 2 )3
Solving ∂ΠE∗ /∂γ ≥ 0, we obtain 196−903γ−342γ 2 +4374γ 3 ≥ 0 for all γ ≥ γ2 ≃ −0.50553.
Thus, ∂ΠE∗ /∂γ > 0. Solving ∂ΠL∗ /∂γ ≤ 0, we obtain −5488 + 39648γ − 137277γ 2 +138024γ 3 > 0 for all γ > γ3 ≃ 0.64468. Thus, ∂ΠL∗ /∂γ < 0. Q.E.D.
B. Deriving equilibrium values in Case 1
Third stage: The producer’s proﬁt in country i yields qii (xii , xji , g) = (1/3)(a − c + g +2xii − xji ) and qij (xjj , xij , g) = (1/3)(a − c − 2g − xjj + 2xij ), where i ̸= j.
Second stage: The maximization problem of the shipper is maxg (g/3)[2(a − c) − 4g −(xHH + xF F ) + 2(xHF + xF H )]. This yields the following shipping fee in the second stage:g(xHH , xHF , xF F , xF H ) = (1/8)[2(a − c) − (xHH + xF F ) + 2(xHF + xF H )].
First stage: Note that Πi is divided into two parts: domestic supply Πii and export
Πij . From the results of the second stage, we obtain the following:[10(a − c) − xjj − 6xji + 15xii + 2xij ]2 − 288γ(xii )2
Πii (xii , xij , xji , xjj ) =,576[2(a − c) − 3xjj − 2xji + 6xij + xii ]2 − 72γ(xij )2
Πij (xii , xij , xji , xjj ) =.
144
From the above equations, FOCs are as follows: ∂Πii /∂xii = 0 ⇔ (5/96)[10(a − c) − xjj −6xji +2xij +15xii ]−γxii = 0 and ∂Πij /∂xij = 0 ⇔ 2(a−c)−3xjj −2xji +6xij +xii −12γxij =0.23 From these equations, we obtain (12).
C. Deriving equilibrium values in Case 2
Third stage: The producer’s proﬁt in country i yields qii (xi , xj , g) = (1/3)(a − c + g +2xi − xj ) and qij (xi , xj , g) = (1/3)(a − c − 2g + 2xi − xj ), where i ̸= j.
23

The SOCs for proﬁt maximization are satisﬁed when the following conditions hold: ∂ 2 Πii /∂x2ii < 0 is
equivalent to γ > 450/576 ≃ 0.78125 and ∂ 2 Πij /∂x2ij < 0 is equivalent to γ > 1/2.

17

Second stage: The maximization problem of the shipper is maxg (g/3)[2(a − c) − 4g +xH + xF ]. This yields g(xH , xF ) = (1/8)[2(a − c) + xH + xF ].
First stage: From the results of the second stage, we obtain the following FOC for the
proﬁt maximization of the producers: (1/288)[226(a − c) + (485 − 288γ)xi − 259xj ] = 0.24
From this, we obtain (14).
D. The eﬀects of γ on equilibrium values
Case 1: The diﬀerentiation of (12) yields
D
∂qij
∂qiiD
4(a − c)(20 − 100γ + 159γ 2 )4(a − c)(−25 + 40γ + 18γ 2 )
<
0;
> 0,
=−
=
∂γ
(10 − 51γ + 48γ 2 )2∂γ
(10 − 51γ + 48γ 2 )2∂xD
∂g D
20(a − c)(13 − 48γ + 60γ 2 )6(a − c)(−25 + 40γ + 18γ 2 )
ii
<
0;
> 0,
=−
=
∂γ
(10 − 51γ + 48γ 2 )2∂γ
(10 − 51γ + 48γ 2 )2
∂π D
48(a − c)2 γ(125 − 300γ + 70γ 2 + 72γ 3 )
S
=
> 0,
∂γ
(10 − 51γ + 48γ 2 )3(a − c)2 (2000 − 9560γ + 20670γ 2 − 26371γ 3 + 15504γ 4 )
∂ΠiD
=−
< 0.
∂γ
2(10 − 51γ + 48γ 2 )3Case 2: The diﬀerentiation of (14) yields
C∂qij
6780(a − c)2712(a − c)16272(a − c)
∂qiiC
∂xC
i
=−
=
−
=−
<
0;
<
0;
< 0,
2
2
∂γ
(144γ − 113)∂γ
(144γ − 113)∂γ
(144γ − 113)2
∂π C
∂g C
4068(a − c)390528(a − c)γ
S
=−
<
0;
=−
< 0.
2
∂γ
(144γ − 113)∂γ
(144γ − 113)3

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The SOC is γ > 288/485 ≃ 0.593814.

18

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21

x*L Ha-cL

x E* Ha-cL

1.0
0.04

0.8

0.03

0.6

0.02

0.4

0.01

0.2

Γ`
1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Γ
2

4

6

8

Panel (b): Graph of x∗L

Panel (a): Graph of x∗E

Figure 1: Equilibrium investment level of producers.

22

10

Γ

ΠS* Ha-cL2

f * Ha-cL

0.04

0.25

0.20
0.03

0.15

0.02

0.10

0.01

0.05

2

4

6

8

Γ

0.00

2

4

6

Panel (b): Graph of πS∗

Panel (a): Graph of f ∗Figure 2: Equilibrium shipping fee and proﬁt of the shipper.

23

8

Γ

PiC Ha-cL2

PiC Ha-cL2

0.2015340.20

0.201532

0.2015300.15
0.2015280.10

0.201526

0.201524

0.05

ΓC
5

10

15

20

25

30

ΓC

0.201522

35

Γ
26

28

30

32

34

36

38

Panel (b): Graph of ΠiC (25 < γ < 38).

Panel (a): Graph of ΠiC(12769/8352 < γ < 38).

Figure 3: Equilibrium proﬁt of the producers in two-way trade (Case 2).

24Γ