Solving a hold-up problem may harm all firms:
Downstream R&D and transport-price contracts
Kazuhiro Takauchi
Tomomichi Mizuno

March 2017
Discussion Paper No.1707

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANSolving a hold-up problem may harm all firms:
downstream R&D and transport-price contracts
Kazuhiro Takauchi∗
Faculty of Business and Commerce, Kansai University
Tomomichi Mizuno†
Graduate School of Economics, Kobe University
This version: March 13, 2017
Abstract
In vertical relations, by raising input price after downstream research and development(R&D) investment, upstream firms can extract the R&D benefit and have an incentive
to set higher input price. As downstream firms underinvest for fear of this hold-up
by upstream firms, outputs and input-demand shrink, and all firms become worse oﬀ.
Previous literature emphasizes that a fixed-price contract in which upstream firms
first commit themselves to input prices and downstream firms subsequently invest canresolve the hold-up problem and make all firms better oﬀ. By contrast, we show that ina vertical relation between firm-specific carriers and exporters, the fixed-price contractof transport price can make all firms worse oﬀ because an eﬃciency improvement inexporters intensifies inter-regional competition. We also discuss the robustness of theresult.
Key words: Transport-price contracts; Downstream R&D; Firm-specific carrier; Holdup problem
JEL classification: L13; F12; O31; R40∗

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: +81-6-6339-7704.
†
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; Tel.: +81-78-803-7245; Fax: +81-78-803-7289.

1

Introduction

In vertical relations, research and development (R&D) investments by downstream
firms give upstream agents an incentive for opportunistic behavior. Suppose that, forexample, a downstream firm invests to reduce its production cost. After observing theinvestment activity, by setting higher input-price, the upstream trading partner can
extract the R&D benefit from the downstream firm. Some fear that the downstream
investment is held-up by the upstream firm. Additionally, the upstream firm’s hold-up
reduces downstream investment,1 and hence, it tends to decrease both upstream anddownstream firms’ profit. Other studies find that an eﬀective way to overcome thisproblem is to fix the input-price through a long-term price agreement, i.e., a fixed-pricecontract (Banerjee and Lin, 2003; Zikos and Kesavayuth, 2010) because the downstream
firm decides its investment level as input-price given, so it does not mind a profit
reduction from a higher input price.
This hold-up problem also can appear in a vertical relation between exporting andtransporting firms. International transportation is an essential service to ship products
overseas, and an exporting firm pays freight rates to cargo carriers to export. Hence, bysetting a higher price after the exporting firm’s investment, it is possible for the carrier
to extract the R&D benefit.2 Actually, transport cost is a major trade barrier, as much
1
The importance of upstream firms’ opportunistic behavior, which causes downstream firms tounderinvest (i.e., the hold-up problem) is widely recognized among researchers. For example, Gilbert
and Cvsa (2003) give an example of a key-component supplier such as Intel and the investment by a
PC maker such as Dell in the computer industry, and emphasize that this sort of hold-up problem isvery likely to occur in supply chains such that downstream firms depend on their trading partners forboth knowledge and capacity. Banerjee and Lin (2003) also give a similar example from the computerindustry.
2
Hummels et al. (2009) empirically show that transport prices, such as the ocean freight rate, is a
mark-up price and carriers have monopoly power. This empirical evidence implies that carriers possibly1

as or larger than other representative policy barriers3 and aﬀects firms’ innovation
activities. For example, because a higher transport cost limits access to foreign markets
and inhibits export production, it aﬀects incentives to innovate, such as cost-reducing
R&D.4
We consider a hold-up problem in international transportation and show that afixed-price contract that resolves the problem in this transport market has entirely
diﬀerent eﬀects than those found in the existing literature.
Our model is based on a Brander and Krugman (1983)-type two-country duopoly
competition. There are two firm-specific carriers upstream and two exporters downstream. Each country has both a carrier and an exporting firm. Each carrier takes a
per-unit transport charge from its domestic exporting firm and ships products to theforeign market. Each exporting firm pays a transport charge to its country’s carrier inorder to export, while it freely supplies to the domestic market. Suppose that in thismarket structure, exporters can commit to zero exports. Then, each exporting firm isa monopoly firm in its local market and can thus gain maximum profit. We expectthat a commitment to fewer exports benefits exporters.
If there is the hold-up problem, exporter will have a high marginal cost because
their investment becomes small. Hence, the hold-up problem is equal to a commitment
to fewer exports. At the same time, the marginal cost of domestic production also

extract rent from exporters thorough price setting.
3
According to Anderson and Van Wincoop (2004), the ad-valorem tax equivalent of transport costs
is 10.7%, while that of tariﬀ and non-tariﬀ barriers is 7 % in developed nations.
4
Innovation incentives are influenced by such factors as market access and the intensity of competition. Since transport cost aﬀects all of these factors, it ultimately aﬀects producer’s innovation
incentives. See, for example, Aghion et al. (2004, 2005).

2

becomes large, which lowers exporters’ profit. Exporters face this trade-oﬀ and benefit
from the hold-up problem if the commitment eﬀect of fewer exports resulting from thehold-up problem is dominant.
In our analysis, when the fixed-price contract is not employed and exporters cancommit to fewer exports, they can maximize their profit because they can create a situation close to domestic monopoly by adjusting the transport price through their investment decision. By contrast, when the fixed-price contract is employed, the exporter’s
profit is minimized because carriers set a lower price to promote exporters’ investments
and exports become the most active. Furthermore, when the cost-reducing R&D ishighly eﬃcient, carriers profit less compared to other contract schemes because carriers
set considerably lower prices to promote investments. Thus, the market structure is
one in which the fixed-price contract harms all firms. We further discuss the robustnessof our results in two cases: one with a positive spillover in R&D, and the other withhorizontally diﬀerentiated domestic and foreign products. We find that our result holds
in these extended cases.
This study is closely related to Takauchi’s (2015a, b) consideration of R&D rivalry5
with international transportation.6 Takauchi (2015a) examines the eﬀects of the tech-

5

Ghosh and Lim (2013), Haaland and Kind (2008), and Long et al. (2011) also consider the relationship between trade costs and innovation. They examine the eﬀects of reduced exogenous trade-cost
on R&D investment.
6
Other studies also consider an imperfectly competitive transport sector in international trade (e.g.,
Abe et al., 2014; Behrens et al., 2009; Francois and Wooton, 2001; Ishikawa and Tarui, 2015; Matsushima and Takauchi, 2014). Francois and Wooton (2001) and Behrens et al. (2009) examine theroles of transport prices and transporting firms’ market power in general equilibrium settings. Using a
two-country oligopoly model, Abe et al. (2014) examine the eﬀects of emissions tax in the transport
sector, while Ishikawa and Tarui (2015) examine the eﬀects of several trade policies on the transport
market. Matsushima and Takauchi (2014) consider how seaport privatization influences their usage
fees (trade cost) and welfare in an international oligopoly.

3

nical eﬃciency of R&D on exporters’ profit and shows that higher R&D eﬃciency can
reduce their profit. Takauchi (2015b) considers the eﬀects of transport eﬃciency, R&D
spillovers, and transport market competition on investment and welfare. The author
shows that competition in the transport market may harm consumers. Although theseworks share a basic market structure—exporters freely supply to their domestic market while they must pay freight rates for cargo carrier to export—with this study, we
incorporate transport price contracts and examine the eﬀects on profit and welfare.
This study is also related to works on input-price contracts with downstream investment (Banerjee and Lin, 2003; Gilbert and Cvsa, 2003; Kesavayuth and Zikos, 2009;
Zikos and Kesavayuth, 2010). Banerjee and Lin (2003) show that fixed-price contracts
make all firms better oﬀ in a market with an upstream monopoly and a downstream
oligopoly. Zikos and Kesavayuth (2010) confirm that Banerjee and Lin’s result always
holds, even if R&D spillovers exist. Gilbert and Cvsa (2003) consider the role of final
demand uncertainty in a supply-chain with one supplier and one buyer. They showthat the supplier prefers a commitment to wholesale prices if the demand fluctuations
are not too large, and the buyer always profits more when the supplier makes a pricecommitment. Kesavayuth and Zikos (2009) examine the role of R&D spillovers andthe importance of wages for labor unions on an endogenous choice of contract form forwages in a union–firm pair. However, these analyses are all limited to the domestic
market and do not consider international trade and transportation. We believe that
our analysis compliments the existing literature.
The rest of the paper proceeds as follows. Section 2 presents the model. Section

4

3 examines transport-price contracts. We discuss the robustness of the main result in
Section 4. Section 5 concludes. We provide the proofs in the appendix.

2

Model

We consider a two-way trade model with firm-specific carriers, as in Takauchi (2015a,b). Two symmetric countries, H and F , have a homogeneous product market. Eachcountry has a single exporting firm (called firm i, i = H, F ) and a firm-specific cargo
carrier (called carrier i). The inverse market demand in country i is pi = a − qii − qji(i, j = H, F ; j ̸= i), where pi is the product price, qii is firm i’s domestic supply, qji isfirm j’s exports, and a > 0. While firm i freely supplies to the domestic market, it mustuse carrier i and pay a per-unit transport-price, ti , to ship its product to an overseas
market. Before production, firm i invests in R&D to reduce marginal production costc (> 0); after the investment, the marginal cost is c − xi , where xi is the investment
level. We assume that the R&D cost function is γx2i ; γ (> 0) denotes the technicaleﬃciency in R&D.7 Firm i’s profit is given byΠi ≡ (a − qii − qji − (c − xi ))qii + (a − qjj − qij − (c − xi ) − ti )qij − γx2i ,where i, j = H, F ; j ̸= i. Carrier i makes a take-it-or-leave-it oﬀer to firm i and decides
its transport price. Each carrier’s profit is πi ≡ ti qij .8
We consider three transport-price contract schemes. The first is a fixed-price contract where each carrier first sets its transport price and firms subsequently invest. The
7This is a popular setting. See, for example, d’Aprémont and Jacquemin (1988), Ghosh and Lim(2013), Haaland and Kind (2008), and Takauchi (2015a).
8
Our main result does not change, though the other trade cost τ exists (i.e., πi ≡ (ti − τ )qij ).

5second is a floating-price contract where firms first invest and each carrier subsequently
sets its transport price. The third is a simultaneous move scenario where carriers andfirms simultaneously decide transport prices and investment levels. In all schemes, eachfirm decides its output (i.e., domestic supply and exports) in the final stage of the gameand competes à la Cournot in both markets in countries H and F . The game is solved
by backward induction.

3

Results

In the final stage, each firm decides its outputs to maximize its profit. The first-orderconditions (FOCs) for profit maximization are ∂Πi /∂qii = a − c − 2qii − qji + xi = 0 and∂Πi /∂qij = a−c−2qij −qjj +xi −ti = 0. These yield qii (tj , x) = (a−c+tj +2xi −xj )/3
and qij (ti , x) = (a − c − 2ti + 2xi − xj )/3. Let x = (xi , xj ).
Fixed-price contract. In the second stage, firm i chooses an investment level, xi ,taking ti as given. From the firm’s FOC, the second-stage investment level isxi (t) =4(3γ − 4)(a − c) − 4(3γ − 2)ti + 6γtj
, j ̸= i.
(3γ − 4)(9γ − 4)(1)

Let t = (ti , tj ). From the third-stage exports qij (ti , x) and (1), carrier i’s maximization
problem ismaxtiti [9γ(3γ − 4)(a − c) − 2(3γ − 1)(9γ − 8)ti + 8(3γ − 1)tj ].
3(3γ − 4)(9γ − 4)This yields the following equilibrium transport price:tfi x =

9γ(3γ − 4)(a − c).
4(3γ − 1)(9γ − 10)6(2)

The outcome in the fixed-price contract is labeled “f x.” From (2), we have the equilibrium investment and outputs:
xfi x =
qiif x =(189γ 2 − 276γ + 80)(a − c).
2(3γ − 1)(9γ − 10)(9γ − 4)3γ(135γ 2 − 210γ + 64)(a − c)3γ(9γ − 8)(a − c)
fx
; qij
=
,
4(3γ − 1)(9γ − 10)(9γ − 4)2(9γ − 10)(9γ − 4)

(3)
(4)

The carrier’s profit and the firm’s profits are
πif x =

27γ 2 (3γ − 4)(9γ − 8)(a − c)2,8(3γ − 1)(9γ − 10)2 (9γ − 4)γ(190269γ 5 −717336γ 4 +1024488γ 3 −686592γ 2 +215808γ−25600)(a − c)2
Πfi x =
.
2
2
2

(5)

16(3γ − 1) (9γ − 10) (9γ − 4)

To ensure a positive quantity, we assume the following throughout the analysis.9
Assumption 1. γ > 4/3.
Floating-price contract. In this contract scheme, carrier i decides its transport
price, ti , in the second stage of the game. The carrier’s maximization problem yields
the following second-stage transport price:1
ti (x) = (a − c − xj + 2xi ), j ̸= i.
4(6)From (6) and the firm’s profit, the equilibrium investment level is∂Πi (x)143(a − c)= (43(a−c)−22xj −(144γ−65)xi ) = 0, j ̸= i ⇒ xli =.
∂xi
72
144γ − 43(7)

The outcome in the floating-price contract is labeled “l.” From (7), we get the following

9

As long as Assumption 1 holds, the second-order conditions for the carriers’ and firms’ profit
maximization are satisfied.

7

outcomes:
tli =qiil =πil =36γ(a − c),144γ − 43
60γ(a − c)24γ(a − c)
l
; qij
=
,
144γ − 43144γ − 43864γ 2 (a − c)2γ(4176γ − 1849)(a − c)2
l
;
Π
=
.
i
(144γ − 43)2
(144γ − 43)2

(8)
(9)
(10)

Simultaneous move scenario. In this case, both transport prices and investments
are decided in the first-stage of the game simultaneously. Eqs. (1) and (6) yield thefollowing:
tsi =
xsi =
qiis =πis =9γ(a − c),2(18γ − 7)
7(a − c),18γ − 73γ(a − c)15γ(a − c)
s
; qij
=
,
2(18γ − 7)18γ − 727γ 2 (a − c)2γ(261γ − 196)(a − c)2; Πsi =.
22(18γ − 7)4(18γ − 7)2

(11)
(12)
(13)
(14)

The outcome in the simultaneous move scenario is labeled “s.”From (3), (7), and (12), we obtain Lemma 1.
Lemma 1. (i) xfi x > xsi > xli . (ii) ∂xki /∂γ < 0 for all k ∈ {f x, l, s}.
The logic behind part (i) is as follows. In the floating-price contract, firms first invest
and carriers subsequently charge transport prices. Then, if firms invest a higher amount,carriers set a higher transport price and can extract a benefit from the R&D. Hence,
to keep lower transport price, firms have an incentive to make smaller investments.
According to this strategic motive, the floating-price contract will have the smallest
investment among all of the schemes. By contrast, carriers first charge transport prices

8and firms subsequently invest in the case of a fixed-price contract. Lower transport
prices enhance transport demand and, can thus raise investments. As found in thesecond-stage investment, xi (t), a lower ti increases xi , so carrier i sets a lower transport
price. The strategic motive of carrier i makes the transport price the lowest among allof the schemes. Corresponding to this low transport price, the investment level becomesthe largest. On the one hand, in the simultaneous move scenario, firms cannot directlyreduce transport prices by setting a smaller investment; hence, the investment is larger
than that in the floating-price contract case (xsi > xli ). Carriers also cannot directly
raise investment by setting a lower transport price; thus, the investment is smaller
than that in the fixed-price contract case (xsi < xfi x ). Part (ii) is intuitive. A smaller
γ improves eﬃciency in R&D and strengthens innovation incentives. The investment
rises as γ decreases (see Fig. 1).

xikHa-cL
0.5
0.4
0.3
0.2

1.5

2.0

2.5

3.0

Γ

Figure 1: R&D investment
Note: Black line is k = f x; Gray line is k = l; Dashed line is k = s.

Eqs. (2), (8), and (11) yield Lemma 2.

9Lemma 2. (i) tsi > tli > tfi x . (ii) If γ > 2
γ<2

(√
)/
15 + 5 3 ≃ 5.91532, ∂tfi x /∂γ < 0, and if)/(√
15 + 5 3, ∂tfi x /∂γ > 0. ∂tli /∂γ < 0 and ∂tsi /∂γ < 0.

Part (i) of Lemma 2 has the following intuitive explanation. In the fixed-pricecontract, carriers commit to lower transport prices in order to increase investment, andtherefore, the price in that scheme becomes the lowest. In the floating-price contractand simultaneous move scenario, the firms’ investment level is a given for carriers.
In these schemes, if a realized investment becomes larger, firms’ production expands,
incentivizing carriers to set a higher price. As shown in Lemma 1, investment in thesimultaneous move scenario is larger than that in the floating-price contract, so thetransport price in the simultaneous move scenario is higher than that in the floatingprice contract.
We next consider part (ii) of Lemma 2. As in part (ii) of Lemma 1, a smaller γ enhances investment incentives. In the fixed-price contract, carriers can raise investmentsby committing to lower transport prices. If carriers reduce transport prices when γ issmall enough, and hence firms’ investment incentives are large enough, it is possible
to further increase investments. Hence, when γ falls below a certain level, the transport price also decreases. By contrast, in a floating-price contract and simultaneous
move scenario, carriers have no transport price commitment. An increase in investment
due to a lower γ promotes production activities, so the transport price increases as γ
decreases. Fig. 2 illustrates Lemma 2.
Eqs. (4), (9), and (13) yield Lemma 3.
fx
s > q l ; for domestic supply: if γ >Lemma 3. (i) For exports: qij
> qij
ij

10

(√
23521 +

tikHa-cL
0.35
0.30
0.25
0.20
0.15
0.10
0.05
2

3

4

5

6

7

Γ

Figure 2: Transport prices
Note: Black line is k = f x; Gray line is k = l; Dashed line is k = s.

239

)/
(√
)/
225 ≃ 1.74385, qiis > qiif x > qiil and if γ <23521 + 239 225, qiis > qiil > qiif x .

k /∂γ < 0 for all k ∈ {f x, l, s}. Comparative(ii) Comparative statics of exports: ∂qij

statics of domestic supply: if γ < γ̂ ≃ 1.48449, ∂qiif x /∂γ > 0 and if γ > γ̂, ∂qiif x /∂γ < 0.
∂qiil /∂γ < 0 and ∂qiis /∂γ < 0.
We first consider the ranking in output. Although a lower transport price promotess > q l hold. These results depend
exports and impedes domestic supply, tsi > tli and qij
ij

on an investment ranking of xsi > xli . In the floating-price contract, firms commit
to a smaller investment. This commitment lowers the degree of the production cost
reduction, so it does not suﬃciently promote the whole production, and hence leads tos > q l and q s > q l . Comparing the simultaneous move scenario with the fixed-price
qij
ij
ii
ii
fx
s and q s > q f x . This corresponds to the fact that the transport pricecontract, qij
> qij
ii
ii

is the highest in the simultaneous move scenario and is the lowest in the fixed-pricecontract among all schemes, tsi > tli > tfi x (part (i) of Lemma 1). Carriers commit to11

a lower transport price in the fixed-price contract. The carrier’s commitment lowers
the trade barrier and promotes exports. Because competition in the domestic market
becomes more intense, domestic supply decreases. On the one hand, the simultaneous
move scenario will have a higher transport price, which impedes exports and promotesfxs > q l , and thedomestic supply. From these arguments, the export ranking is qij
> qij
ij

domestic supply ranking is qiis > qiil and qiis > qiif x .
The domestic supply ranking between the fixed-price and floating-price contracts
and part (ii) of Lemma 3 are explained as follows. A smaller γ enhances firm’s innovation incentives and raises investment (part (ii) of Lemma 1). Because a decrease in γ
leads to a reduction in production cost, a decrease in γ increases both domestic supply
and exports. However, if γ is small, the domestic supply in the fixed-price contract decreases as γ decreases. This result depends on a change in the transport price for γ. As
in part (ii) of Lemma 2, when γ is small, the transport price in the fixed-price contract
falls as γ decreases. A lower transport price promotes exports, increases competition in
the domestic market, and decreases domestic supply. Hence, ∂qiif x /∂γ > 0 if γ is small.
Furthermore, while a lower γ reduces the domestic supply in the fixed-price contract,
it increases the domestic supply in the floating-price contract. Therefore, if γ is smallenough, the domestic supply in the fixed-price contract can be smaller than that in the
floating-price contract. Fig. 3 illustrates Lemma 3.
Comparing (5), (10), and (14), we establish Proposition 1.
Proposition 1. (i) Πli > Πsi > Πfi x . (ii) If γ < γ ∗ ≃ 1.74661, πis > πil > πif x , and if
γ > γ ∗ , πis > πif x > πil .

12kHa-cLqijkHa-cL
qii

0.50

0.58

0.45

0.56

0.40

0.54
0.35

0.52

0.30

0.50

0.25

0.48

0.20
1.5

2.0

2.5

3.0

Γ

1.5

2.0

2.5

3.0

Γ

Figure 3: Graph of outputs (Exports on the left; domestic supply on the right)
Note: Black line is k = f x; Gray line is k = l; Dashed line is k = s.

This result implies that a better outcome for both carriers and firms can appear
when carriers do not commit to a transport price level. The ranking in firm’s profit(part (i) of Proposition 1) inversely corresponds to the ranking in exports (see part (i)of Lemma 3). This is because, in our two-way duopoly model, the prohibitive transport
price level gives the highest profit for firms. That is, the profit in a domestic monopoly(there is no export) is higher than the situation with aggressive exports.10 When firms
can commit to an investment level, it is possible for them to create a situation close
to a domestic monopoly because they can adjust the transport price through theirinvestment decision. Hence, the floating-price contract is the best scheme for firms. In
contrast, the carrier’s price commitment adjusts the firm’s investments and exports.
The carrier’s lower price commitment increases the aggressiveness of the firm’s export
activities, and it puts firms in a situation furthermost from a domestic monopoly. The
fixed-price contract is the worst for firms. In the simultaneous move scenario, carriers

10

Using the third-stage outcomes and conditions xj = xi and tj = ti , this fact is immediately found.

13and firms have no commitment, so the profit in that scheme is intermediate for firms.
The carrier’s profit ranking (part (ii) of Proposition 1) depends on the transport
price and export volume. In the simultaneous move scenario, the transport price ishighest among all contract schemes and the exports are of intermediate size, so the
profit becomes the largest among all schemes. On the one hand, profits in fixed-priceand floating-price contracts can be reversed according to the degree of γ. This isbecause, as in part (ii) of Lemma 3, a smaller γ increases exports in both schemes
and raises the transport price in the floating-price contract, while it can reduce thetransport price in the fixed-price contract (see also Fig. 2). Then, a smaller γ increases
profit in the floating-price contract. In contrast, a smaller γ decreases profit in the
fixed-price contract when γ is small. If γ is small enough, (i.e., γ < γ ∗ ), profit in the
fixed-price contract is the worst among all contract schemes.

PkiHa-cL2

ΠkiHa-cL2

0.22

0.08

0.21

0.07
0.06

0.20

0.05

0.19

0.04

0.18

0.03

0.17

0.02

1.5

2.0

2.5

3.0

Γ

1.5

2.0

2.5

3.0

Γ

Figure 4: Graph of profits (firm’s profit on the left; carrier’s profit on the right)
Note: Black line is k = f x; Gray line is k = l; Dashed line is k = s.

Welfare analysis. Here, we consider welfare in all contract schemes. Country i’sk )2 /2, where
consumer surplus in each contract scheme is given by CSik = (qiik + qji

14k ∈ {f x, l, s}. This yields the following:
CS f x =9γ 2 (189γ 2 − 276γ + 80)2 (a − c)23528γ 2 (a − c)2
l
;
CS
=
,
32(9γ − 4)2 (9γ − 10)2 (3γ − 1)2(144γ − 43)2

441γ 2 (a − c)2.
CS =8(18γ − 7)2

(15)

s

Each country’s social surplus consists of consumer surplus CSik , firm’s profit Πki ,and carrier’s profit πik . The social surplus in each contract scheme isWif x =

γ(86751γ 4 − 251424γ 3 + 249120γ 2 − 96960γ + 12800)(a − c)2,32(9γ − 10)2 (3γ − 1)2 (9γ − 4)γ(8568γ − 1849)(a − c)2
;
Wil =
2
(144γ − 43)

7γ(153γ − 56)(a − c)2
Wis =
.
2

(16)

8(18γ − 7)

Eqs. (15) and (16) yield the following result.
Proposition 2. (i) CSif x > CSis > CSil and Wif x > Wis > Wil . (ii) ∂CSik /∂γ < 0and ∂Wik /∂γ < 0.
The welfare ranking corresponds to the investment ranking (see part (i) of Lemma1). When firms increase their investment level, the marginal production cost falls andproduction eﬃciencies improve, which undoubtedly increases total sales and consumer
surplus. Because the improvement in consumer benefit is dominant, the social surplus
also increases. From the welfare point of view, the best scheme is the fixed-pricecontract in which investment is maximized, and the worst scheme is the floating-pricecontract in which investment is minimized. Furthermore, a reduction in γ enhanceseﬃciencies in R&D and raises investment, so it also increases welfares.

15

4

Discussion

We have two robustness checks on the main result (Proposition 1). We first relax
the assumption of no R&D spillover and examine a case with a positive spillover.
Subsequently, we argue a case containing domestic and foreign product diﬀerentiation.

4.1

R&D spillovers

Here, we introduce an exogenous spillover rate of R&D, δ, in the previous setting. Since
the developed-knowledge of the rival firm is transmitted, firm i’s marginal production
cost is rewritten as c − xi − δxj (i ̸= j).
√
We assume that the spillover rate is not very large: 0 ≤ δ < (73 − 3 377)/44 (≈0.33529). We exclude the case with a high spillover rate for the following reason. In theprevious section, R&D investments are strategic substitutes. However, under a rather
high spillover rate, the investment decisions become strategic complements. Hence,
under a high spillover rate, we do not have a mechanism in the previous section andwe obtain a significantly diﬀerent profit ranking. Hence, we omit the case with a highspillover rate.
Firm i’s profit is Πi (δ) = (a − qii − qji − (c − xi − δxj ))qii + (a − qjj − qij − (c −xi − δxj ) − ti )qij − γx2i and carrier i’s profit is πi (δ) = ti qij . We consider the same
contract schemes and the same timing of the game. The equilibrium outcomes under16

the fixed-price contract are
xfi x (δ) =(a − c)(δ − 2)[−189γ 2 + 6γ(10δ 2 − 43δ + 46) + 20(δ − 2)2 (δ 2 − 1)],(9γ + 2δ 2 − 2δ − 4)[108γ 2 − 39γ(δ − 2)2 − 10(δ − 2)2 (δ 2 − 1)]

qiif x (δ) =3γ(a − c)[135γ 2 − 3γ(16δ 2 − 67δ + 70) − 16(δ − 2)2 (δ 2 − 1)],(9γ + 2δ 2 − 2δ − 4)[108γ 2 − 39γ(δ − 2)2 − 10(δ − 2)2 (δ 2 − 1)]

fx
qij
(δ) =

3γ(a − c)[54γ 2 − 3γ(4δ 2 − 19δ + 22) − 4(δ − 2)2 (δ 2 − 1)],(9γ + 2δ 2 − 2δ − 4)[108γ 2 − 39γ(δ − 2)2 − 10(δ − 2)2 (δ 2 − 1)]

tfi x (δ) =9γ(a − c)(3γ − 2δ 2 + 6δ − 4),108γ 2 − 39γ(δ − 2)2 − 10(δ − 2)2 (δ 2 − 1)
2

2

2

fx
Πfi x (δ) = [qiif x (δ)] + [qij(δ)] − γ[xfi x (δ)] ;fx
πif x (δ) = tfi x (δ)qij(δ).

The equilibrium outcomes under the floating-price contract are
xli (δ) =(a − c)(43 − 14δ);144γ + 14δ 2 − 29δ − 43

qiil (δ) =60γ(a − c),144γ + 14δ 2 − 29δ − 43

l
qij
(δ) =

24γ(a − c);144γ + 14δ 2 − 29δ − 43

tli (δ) =36γ(a − c),144γ + 14δ 2 − 29δ − 43

2

2

2

l
Πli (δ) = [qiil (δ)] + [qij(δ)] − γ[xli (δ)] ;lπil (δ) = tli (δ)qij(δ).

The equilibrium outcomes under the simultaneous move scenario are
xsi (δ) =7(a − c)(2 − δ);
36γ + 7 (δ 2 − δ − 2)qiis (δ) =15γ(a − c),36γ + 7 (δ 2 − δ − 2)

s
qij
(δ) =

6γ(a − c);36γ + 7 (δ 2 − δ − 2)tsi (δ) =9γ(a − c),36γ + 7 (δ 2 − δ − 2)sΠsi (δ) = [qiis (δ)]2 + [qij(δ)]2 − γ[xsi (δ)]2 ;sπis (δ) = tsi (δ)qij(δ).

Now, we can check the robustness of Proposition 1. Under our assumptions, 0 ≤√δ < (73 − 3 377)/44 and γ > 4/3, we compare equilibrium profit. By numericalcalculation, we have Πli (δ) > Πsi (δ) > Πfi x (δ). Thus, part (i) of Proposition 1 does notchange.
We next consider the profit ranking of carrier i. Numerical calculation yields Fig.
17

5. In the lower left area, we have πis (δ) > πil (δ) > πif x (δ); in the other area, we obtain
πis (δ) > πif x (δ) > πil (δ). Hence, part (ii) of Proposition 1 does not change if the spillover
rate is small. On the other hand, a high spillover rate yields only the profit ranking
πis (δ) > πif x (δ) > πil (δ).

γ
2.0
1.9

πis (δ) > πif x (δ) > πil (δ)
1.81.7πis (δ) > πil (δ) > πif x (δ)

1.6
1.5
1.4

0.00

0.05

0.10

0.15

0.20

0.25

0.30

δ

Figure 5: Profit ranking for carrier i with spillover

Our results change for the following reason. Under a positive spillover rate, anincrease in R&D investment reduces the rival firm’s marginal cost. Then, firms havea small incentive to invest. This case is similar to that with ineﬃcient investment
technology (i.e., large γ). In other words, the eﬀect of an increase in δ is similar thatof an increase in γ. In our model, the investment level plays an important role so wetherefore obtain Fig. 5.

18

4.2

Diﬀerentiated products

We consider the eﬀects of product diﬀerentiation here. To exclude the eﬀect of market
expansion by product diﬀerentiation (Singh and Vives, 1984), we employ the Shubik(1980)-type utility function:11
(
)
( 2
) β
2
2
ui = a (qii + qji ) − (1 − β) qii + qji + (qii + qji ) , j ̸= i.
2β ∈ [0, 1] is interpreted as the degree of product diﬀerentiation. That is, products arehomogeneous at β = 1 and are independent at β = 0. Moreover, under this utility
function, the aggregate demand in country H or F , qii + qji , does not depend on the
degree of product diﬀerentiation. In particular, we have qii + qji = a − (pii + pji )/2.
Hence, we can exclude the market expansion eﬀect.
Solving the utility maximization problem, we have the inverse demand pii = a −(2 − β)qii − βqji and pij = a − (2 − β)qij − βqjj . Then, firm i’s profit is Πi (β) =(a − (2 − β)qii − βqji − (c − xi ))qii + (a − (2 − β)qij − βqjj − (c − xi ) − ti )qij − γx2i andcarrier i’s profit is πi (β) = ti qij . We consider the same contract schemes and the sametiming of the game. The equilibrium outcomes under the fixed-price contract are(a − c)(16 − 16β + 3β 2 )Φ4,
4(2 − β)Φ1 Φ2

xfi x (β) =(a − c)(2 − β)Φ3;2Φ1 Φ2

fx
qij
(β) =

(a − c)(16 − 16β + 3β 2 )γΦ5,
2Φ1 [9β 4 γ + β 3 (6 − 96γ) + 8β 2 (−5 + 44γ)β(88 − 512γ) + 64(−1 + 4γ)]

qiif x (β) =2(a − c)(16 − 16β + 3β 2 ) γΦ6,4(2 − β)Φ2
(
)
2
2
2
fx
Πfi x (β) = (2 − β) [qiif x (β)] + [qij(β)] − γ[xfi x (β)] ;tfi x (β) =11fx
πif x (β) = tfi x (β)qij(β),

Our qualitative results do not change when using the utility function proposed by Singh and Vives(1984).

19

where we define Φm (m = 1, . . . , 6) as in the appendix.
The equilibrium outcomes under the floating-price contract are
xli (β) =(a−c)(320−448β+200β 2 −29β 3 )4(a−c)(8−3β)(16−16β+3β 2 )γ
; qiil (β) =
,
Ψ1
Ψ1
2

4(a − c)(16 − 16β + 3β 2 ) γ8(a − c)(2 − β)(16 − 16β + 3β 2 )γ
; tli (β) =
,
Ψ1
Ψ1
(
)
2
2
2
l
l
Πli (β) = (2 − β) [qiil (β)] + [qij(β)] − γ[xli (β)] ; πil (β) = tli (β)qij
(β),

l
qij
(β) =

where Ψ1 ≡ 48β 4 γ + β 3 (29 − 544γ) + 8β 2 (272γ − 25) − 448β(8γ − 1) + 64(32γ − 5).
The equilibrium outcomes under the simultaneous move scenario are
xsi (β) =(a − c)(2 − β)(12 − 5β);Ψ2qiis (β) =(a − c)(8 − 3β)(16 − 16β + 3β 2 )γ,2(2 − β)Ψ2

(a − c)(4 − 3β)2 (−4 + β)2 γ(a − c)(16 − 16β + 3β 2 )γ
; tsi (β) =,Ψ22(2 − β)Ψ2()
ssΠsi (β) = (2 − β) [qiis (β)]2 + [qij(β)]2 − γ[xsi (β)]2 ; πis (β) = tsi (β)qij
(β),
s
qij
(β) =

where Ψ2 ≡ −6β 3 γ + β 2 (56γ − 5) + β(22 − 160γ) + 8(16γ − 3).
Here, we consider the robustness of Proposition 1. Under our assumptions, 0 ≤ β ≤1 and γ > 4/3, we compare equilibrium profit. By numerical calculation, we depict the
profit ranking of firm i in Fig. 6. On the right, we find Πli (β) > Πsi (β) > Πfi x (β); inthe middle, we have Πli (β) > Πfi x (β) > Πsi (β); and on the left, we obtain Πfi x (β) >Πil (β) > Πsi (β). That is, in the case with higher product diﬀerentiation, the firms with
fixed-price contracts earn larger profits.
The reason is as follows. When products are highly diﬀerentiated, competitions
between firms become less important. Then, the firms’ profits depend on vertical
relationships between the firms and carriers. Hence, a commitment toward aggressive

20Πfi x (β) > Πli (β) > Πsi (β)Πli (β) > Πsi (β) > Πfi x (β)

γ
10

8

6

4

2
0.0

0.2

0.4

0.6

0.8

1.0

β

Πli (β) > Πfi x (β) > Πsi (β)
Figure 6: Profit ranking for firm i with product diﬀerentiation
investment will increase profit in a channel. Therefore, product diﬀerentiation brings
higher profits to the firms.
Next, we consider the profit ranking of carrier i. Fig. 7 depicts the numericalcalculation. In the lower right area, we have πis (β) > πil (β) > πif x (β); in the middle,
we find πis (β) > πif x (β) > πil (β); and on the left, we obtain πif x (β) > πis (β) > πil (β).
Hence, under higher product diﬀerentiation, carriers prefer fixed-price contracts. The
intuition behind this result is similar as in the case of firms.

21πif x (β) > πis (β) > πil (β)πis (β) > πif x (β) > πil (β)

γ
3.0

2.5

2.0

1.5

0.0

0.2

0.4

0.6

0.8

1.0 β

πis (β) > πil (β) > πif x (β)
Figure 7: Profit ranking for carrier i with product diﬀerentiation

5

Conclusion

In a vertical production relationship, upstream trading firms likely hold up downstream
R&D investment. If these upstream firms set a higher input price after observing downstream investment, then they can extract the downstream R&D benefit, and such opportunistic behavior reduces downstream innovation incentives. The previous literatureemphasizes that a fixed-price contract for the input price is required to overcome thishold-up problem. In the fixed-price contract, upstream firms commit to an input pricelevel and downstream firms subsequently invest in cost-reducing R&D. By employing
the fixed-price contract, upstream firms set a lower input price to promote downstream

22

investment. Since this lower-price commitment increases outputs and demand for inputs through investment expansion, all firms become better oﬀ.
In contrast to this standard theory, we show that the fixed-price contract can harm
all firms. We consider a two-country, two-way trade model with two firm-specific carriers upstream, and two exporters downstream. Each country has a carrier and an
exporting firm. While each country’s exporting firm freely supplies to the domestic
market, it uses a local carrier to export its product. Each carrier charges a transport
price and conveys its domestic exporting firm’s product. In this setting, exporters cancreate a situation close to a domestic monopoly when the transport price is high enough.
Although the domestic monopoly is most profitable for exporters, the fixed-price contract lowers the transport price and encourages firms to invest and export aggressively,
creating a market furthest from a domestic monopoly. This makes exporters worse oﬀ.
Furthermore, if R&D eﬃciency is high enough, carriers set considerably low transport
prices in the fixed-price contract. This also makes carriers worse oﬀ. Moreover, we examine the robustness of the main result in two diﬀerent situations: a case with positive
R&D spillover and a case with product diﬀerentiation. We find that in these extended
cases, our main result holds.
This study shows that a fixed-price transportation contract can harm all firms inboth upstream and downstream markets when downstream (exporting) firms engage
in cost-reducing R&D. Therefore, it would be interesting to investigate other formsof R&D, such as product innovation and product quality improvement to determine
whether our result holds. However, this issue is beyond the scope of this study, and we23thus leave it to future research.

Acknowledgments
We thank Noriaki Matsushima for his helpful comments. Any remaining errors are ourown.

Appendix
Proof of Lemma 1. (i) Comparing investments, we get
xfi x −xsi =234γ(a − c)9γ(15γ − 8)(a − c)
> 0; xsi −xli => 0.
2(3γ−1)(9γ−10)(9γ−4)(18γ−7)(18γ−7)(144γ−43)(ii) Diﬀerentiating (3), (7), and (12) w.r.t. γ, we obtain∂xfi x27(1701γ 4 − 4968γ 3 + 5130γ 2 − 2160γ + 320)(a − c)
=−
< 0,
∂γ
2(9γ − 4)2 (9γ − 10)2 (3γ − 1)2
6192(a − c)126(a − c)
∂xli
∂xsi
=−
=−
<
0;
< 0.
2
∂γ
(144γ − 43)∂γ(18γ − 7)2

Q.E.D.

Proof of Lemma 2. (i) Comparing transport prices, we gettsi − tli =117γ(a − c)27γ(27γ − 4)(a − c)
> 0; tli − tfi x => 0.
2(18γ−7)(144γ−43)4(3γ−1)(9γ−10)(144γ−43)(ii) Diﬀerentiating (2), (8), and (11) w.r.t. γ, we have∂tfi x∂tsi−1548(a−c)∂tli−63(a−c)−9(9γ 2 −60γ+40)(a−c)
=
<
0;
=
<
0;
=
.
∂γ
(144γ−43)2∂γ
2(18γ−7)2∂γ
4(9γ−10)2 (3γ−1)2From the last equation, ∂tfi x /∂γ < (≥) 0 if γ > (≤) 2)(√
15 + 5 /3 ≃ 5.91532. Q.E.D.

Proof of Lemma 3. (i) Comparing exports, we get
fx
s
qij
− qij
=

9γ(15γ − 8)(a − c)39γ(a − c)sl
> 0; qij
− qij
=
> 0.
2(9γ − 10)(9γ − 4)(18γ − 7)(18γ − 7)(144γ − 43)24Comparing domestic supplies, we have195γ(a−c)9γ(3γ−2)(15γ−8)(a−c)
qiis − qiil = 2(18γ−7)(144γ−43)> 0; qiis − qiif x = 4(3γ−1)(9γ−10)(9γ−4)(18γ−7)> 0,3γ(675γ −1434γ+448)(a−c)
qiif x − qiil = 4(3γ−1)(9γ−10)(9γ−4)(144γ−43).
2From qiif x − qiil , solving 675γ 2 − 1434γ + 448 ≥ 0 w.r.t. γ, we have qiif x < (≥) qiil if γ <√(≥) ( 23521 + 239)/225 ≃ 1.74385. (ii) From (4), (9), and (13), we get
fx

s
l
∂qij
∂qij
∂qij
2 −360γ+160)(a−c)−1032(a − c)−21(a − c)
=
<
0;
=
< 0;
= −3(243γ< 0,
2 (9γ−10)2(9γ−4)
2
2∂γ(144γ − 43)∂γ(18γ − 7)
∂γ

∂qiil
∂qiis
= −2580(a−c)= −105(a−c)< 0,
2 < 0;(144γ−43)2(18γ−7)2∂γ∂γ
4
3 +38484γ 2 −16800γ+2560)(a−c)∂qiif x= −3(10935γ −35316γ.
4(3γ−1)2 (9γ−10)2 (9γ−4)2∂γ

From the last equation, ∂qiif x /∂γ > (≤) 0 if γ < (≥) γ̂ ≃ 1.48449. Q.E.D.
Proof of Proposition 1. (i) Comparing the firm’s profits, we getΠli − Πsi =117γ 2 (5904γ − 1945)(a − c)2> 0,4(18γ − 7)2 (144γ − 43)2

−12879γ −864γ +4440γ−1024)(a−c)Πsi − Πfi x = 9γ (15γ−8)(8748γ> 0.
16(3γ−1)2 (9γ−10)2 (9γ−4)2 (18γ−7)2
2

4

3

2

2

(ii) Comparing the carrier’s profits, we have
2

2

2

2

(32γ−11)(a−c)
27γ (15γ−8)(27γ−4)(a−c)πis − πil = 3159γ> 0, πis − πif x = 8(3γ−1)(9γ−10)2 (9γ−4)(18γ−7)2 > 0,2(18γ−7)2 (144γ−43)2−285309γ +214692γ−43232)(a−c)
and πif x − πil = 27γ (101088γ. From the last equation, we8(3γ−1)(9γ−10)2 (9γ−4)(144γ−43)2
2

3

2

2

find that πif x − πil ≤ (>) 0 if γ ≤ (>) γ ∗ ≃ 1.74661. These imply Proposition 1. Q.E.D.
Proof of Proposition 2. (i) From (15), we have
CSif x − CSis =81γ 3 (15γ − 8)(6804γ 3 − 12717γ 2 + 6816γ − 1120)(a − c)2>032(3γ − 1)2 (9γ − 10)2 (9γ − 4)2 (18γ − 7)2

25

2

2

(32γ−11)(a−c)> 0. From (16), we haveand CSis − CSil = 51597γ
8(18γ−7)2 (144γ−43)2

Wif x − Wis =9γ 2 (15γ − 8)(4860γ 3 − 8667γ 2 + 4056γ − 560)(a − c)2>032(3γ − 1)2 (9γ − 10)2 (9γ − 4)(18γ − 7)22

2(5760γ−2149)(a−c)
and Wis − Wil = 117γ
> 0.
8(18γ−7)2 (144γ−43)2(ii) Diﬀerentiating eqs. (15) amd (16) w.r.t. γ, we get2
4 −54108γ 3 +53856γ 2 −22080γ+3200)(a−c)2∂CSif x= − 9γ(189γ −276γ+80)(19683γ< 0,
3 (9γ−10)3 (9γ−4)316(3γ−1)∂γ∂CSil303408γ(a − c)2∂CSis3087γ(a − c)2
=−
<
0,
=
−
< 0,
∂γ
(144γ − 43)3∂γ
4(18γ − 7)3)(
4586139γ 6 −19503666γ 5 +33045084γ 4 −28257120γ 3∂Wif x∂γ=−+12744000γ 2 −2880000γ+256000
16(9γ − 4)2 (3γ − 1)3 (9γ − 10)3

(a − c)2< 0,43(10944γ−1849)(a−c)249(81γ−28)(a−c)2
∂Wil
∂Wis
=−
=
−
<
0,
and
< 0.
∂γ
(144γ − 43)3∂γ
4(18γ − 7)3
Q.E.D.
Definition of Φm . We define Φm (m = 1, . . . , 6) as follows: Φ1 ≡ 3β 3 γ + β 2 (4 − 28γ) +16β(5γ − 1) − 64γ + 16, Φ2 ≡ 27β 6 γ 2 + 27β 5 (1 − 16γ)γ +6β 4 (1 − 55γ + 456γ 2 ) − 4β 3 (13 −386γ +2176γ 2 )+24β 2 (7−144γ +608γ 2 )−16β(15−232γ +768γ 2 )+128(1−12γ +32γ 2 ),
Φ3 ≡ 135β 7 γ 2 − 36β 6 γ(69γ − 5) + 48β 5 (1 − 53γ + 393γ 2 ) − 16β 4 (32 − 909γ + 4772γ 2 ) +128β 3 (17−337γ+1386γ 2 )−512β 2 (9−137γ+462γ 2 )+256β(19−232γ+656γ 2 )−2048(1−10γ +24γ 2 ), Φ4 ≡ 81β 7 γ 2 −6β 6 γ(252γ −19)+8β 5 (4−205γ +1458γ 2 )−8β 4 (44−1193γ +6000γ 2 )+128β 3 (12−225γ +886γ 2 )−256β 2 (13−186γ +600γ 2 )+512β(7−80γ +216γ 2 )−512(3−28γ+64γ 2 ), Φ5 ≡ −9β 4 γ−β 3 (8−96γ)−16β 2 (22γ−3)−β(96−512γ)−64(4γ−1),and Φ6 ≡ −9β 3 γ − β 2 (4 − 60γ) − 16β(7γ − 1) + 64γ − 16.

26

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