Consumer-benefiting transport cost: The role of productinnovation in a vertical structure
Kazuhiro Takauchi
Tomomichi Mizuno
September 23, 2020
Discussion Paper No.2017
Revised No.1911

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANConsumer-benefiting transport cost: The role of productinnovation in a vertical structure
Kazuhiro Takauchi∗† and Tomomichi Mizuno‡
This version: September 23, 2020
Abstract
We study the eﬀects of a reduction of transport cost on the firm’s activity of product innovationand on consumer welfare. Firms engage in product R&D that increases the degree of productdiﬀerentiation, purchase intermediate inputs from an exclusive supplier, and export their products to the foreign market paying a per-unit transport cost. Trade theory commonly assertsthat zero transport cost maximizes consumer surplus. Contrary to this standard belief, we showthat a positive transport cost can maximize consumer surplus. We also consider more generaleﬀects of R&D and show that our main results hold even in such case.

Key words: Transport cost; Consumer surplus; Product R&D
JEL classification: L13; F12; O31∗

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.
†
Research Fellow, Graduate School of Economics, Kobe University, 2-1 Rokkodai-cho, Nada-ku, Kobe-City,
Hyogo 657-8501, Japan.
‡
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

Trade barriers such as transportation cost and tariﬀs aﬀect consumer welfare and the incentive offirms to innovate. When market access to foreign firms is facilitated, inexpensive imported goods
become available to domestic consumers, which benefits them.1 Furthermore, if a reduction inshipping costs facilitates the access to the foreign market, then exporting firms can enjoy a larger
market size. The growth in the market size implies that a firm’s sales opportunities increase.
Then, the possibility of increasing production capability can increase the incentive to invest in
research and development (R&D).2
Trade theory commonly argues that reducing transport cost, for example, through tradeliberalization, makes consumers better oﬀ. In the case of intra-industry trade, Helpman andKrugman (1985, p. 108) state: “Trade has a procompetitive eﬀect: each firm’s exports do notdisplace an equal volume of shipments from the other firm to its home market, so total output
and consumption rise and the price falls.” According to this argument, a decline in transport
cost facilitates exports and, thereby, increases total output while reducing prices.3 Hence, therationale in standard theory would lead one to expect that zero transport cost, that is, free
trade, maximizes consumer surplus.
This study examines the eﬀects of transport cost on a firm’s product innovation4 and on
consumer welfare in a two-country, two-way trade model. In each country, the firm exporting
the final good engages in product R&D (which increases the degree of diﬀerentiation of the
product), and purchases input from an exclusive supplier. We show that a lower transport
1

For example, Nicita (2009) empirically finds that Mexican tariﬀ liberalization during the 1990s reduced
consumer prices for agricultural and manufactured goods and raised consumers’ purchasing power in about 1.8%.
2
Some recent empirical works strongly support this view. For example, Lileeva and Trefler (2010) find that
trade liberalization leads firms to engage in more product innovation. Bustos (2011) also finds that regional tradeliberalization (Brazil’s tariﬀ cut) fosters firms’ innovation (technology upgrading by Argentinian firms).
3
This result holds in other extended cases. For example, Naylor (1998) shows that transport cost reduction
always increases consumer surplus, even if the reciprocal market model incorporates unionized labor markets.
Maiti and Mukherjee (2013) also find this result in a unionized oligopoly model.
4This setting is partially consistent with the empirical evidence. According to a survey by Fontana and Guerzoni (2008) in major manufacturing industries, most firms consider product innovation as their most important
form of innovation.

1

cost leads the final-good exporting firms to invest in product R&D. However, contrary to thestandard belief, we show that a positive transport cost can maximize consumer surplus.
The key assumptions to our results are “strategic complements in R&D” and “input-pricejumps due to investment.” When the R&D cost is too high, no firm will invest in R&D.
However, when the R&D cost is small, the incentive to invest in R&D depends on the transportcost. When the home and foreign exporting firms invest in R&D, they benefit from the productdiﬀerentiation, which moderates competition. However, if the transport cost is high, the market
size in the other country is small, and the benefits of product diﬀerentiation are also small.
Hence, if the transport cost is high, non-investment is the dominant strategy. Conversely, if thetransport cost is low, then the benefit of product diﬀerentiation in the other market is large, andinvestment is the dominant strategy. When the transport cost is at an intermediate level, there
is no dominant strategy, and multiple equilibria, the “everyone invests” and “no one invests”
regimes, appear.
In our model, each exporting firm uses its domestic supplier’s input. R&D activities takea much longer time, so exporting firms first invest in R&D, then input suppliers, subsequently,
charge their prices. In this case, by charging a higher price after observing the exporting firm’s
investment behavior, the input suppliers can extract R&D benefits. Hence, the investment by
the exporting firm not only increases the input price, but also the production cost. Due to thesehigher costs, each firm’s production falls and total output decreases, so the final-good pricerises. Therefore, R&D investment reduces consumer surplus. Furthermore, since the strategic
complements eﬀect influences input prices, the home input supplier raises its price when the othercountry’s input supplier raises its price. As the number of investing firms increases, input prices
rise and the total output decreases, so consumer surplus also drops as the number of investing
firms increases. The “everyone invests” regime has the lowest consumer surplus level of all theregimes. If the transport cost decreases below a certain level, then consumer surplus drops

2because the equilibrium switches from the “no one invests” regime to the “everyone invests”
regime.
We believe that our analysis can contribute to the field of trade and competition. In competition policy, the consumer welfare standard is often adopted (Viscusi et al. 2018), and its
purpose is to protect consumers’ wellbeing and enhance consumer surplus. Hence, if a reduction
in transport cost increases consumer surplus, the authority in charge of competition policy has
an incentive to decrease transport cost. Because the reduction in transport cost tends to be
seen as beneficial to consumers, it might be actually implemented. However, our result showsthat a reduction in transport cost does not always bring about the desirable result of increased
consumer welfare, and points out to a problematic issue in competition policy.
Whilst our result opposes the conventional wisdom in trade theory, it is not necessarily
inconsistent with empirical findings. Examples of price rises due to the promotion of competition
have been reported by several empirical studies (Caves et al. 1991, Grabowski and Vernon 1992,
and Thomadsen 2007). Transport cost reduction lowers the export barrier to foreign firms andenhances competition in the domestic market, so it undoubtedly has a competition-enhancing
eﬀect. The result that the promotion of competition raises the final-good price, and hence,harms consumers, is broadly consistent with empirical evidence.
This paper is closely related to Kabiraj and Marjit’s (2003) study. By considering technology
transfer through licensing, they show the existence of a tariﬀ rate that maximizes consumer
surplus in an importing country. In their model, a high tariﬀ rate hinders the exports of aforeign firm, so the foreign firm has an incentive to license its low-cost production technology
to the high-cost local firm and earn license royalties. As a result, a high tariﬀ of the importing
country increases total output, thereby increasing the consumer surplus above that of the zero
tariﬀ case. Although Kabiraj and Marjit (2003) find the existence of a consumer-benefiting tariﬀ
policy, their analysis is limited to the case of one-way trade. By contrast, here we examine the3two-way trade case.5
This paper is also related to some works on the nexus between trade liberalization andproduct innovation (Bastos and Straume, 2012; Braun, 2008; Hwang et al., 2018).6 In thecontext of oligopolistic intra-industry trade, Bastos and Straume (2012) and Braun (2008) use a
product R&D model based on Lin and Saggi (2002). However, they focus on the role of skilled
and unskilled workers and consider the eﬀects of trade liberalization on wage inequalities andlabour demands through the impact on firm’s innovation. Hwang et al. (2018) employ a typeof product R&D in which investment aﬀects consumers’ willingness to pay, and show that tradeliberalization may reduce R&D investments.
The rest of the paper proceeds as follows. Section 2 describes the model. In Section 3, wederive the equilibrium outcomes and the main results. In Section 4, we consider a general casewith a smaller degree of product diﬀerentiation due to R&D compared to the case in Section 3.
Section 5 concludes the paper.

2

Model

We consider a reciprocal market as in Brander and Krugman (1983). There are two symmetriccountries, H and F , with final-good markets. Each country i (i = H, F ) has an input supplier(called supplier i) and an innovative final-good exporting firm (called firm i). To produce one
unit of the final good, firms employ one unit of input. We assume that firm i buys its input
from an exclusive supplier i. That is, firm i is technologically unable to use the input producedby (foreign) supplier j (j ̸= i). This is assumed because the intermediate input employed by

5

Furthermore, in contexts diﬀerent from our analysis, some studies examine the eﬀects on consumer welfare
arising from diﬀerent organizational forms among firms and market structure. In a free-entry situation, Marjit andMukherjee (2015) indicate that a transport cost reduction may decrease consumer surplus. Cao and Mukherjee(2017) find that, if a labour union exists and technology transfer is implemented among firms, then consumer
surplus may increase. Using a third-market model with export policy, Mukherjee and Sinha (2019) show thatcooperation between exporting firms harms consumers in the third market.
6
In addition, several studies focus on the nexus between process innovation (i.e., cost-reducing R&D) and tradeliberalization. See, for example, Haaland and Kind (2008) and Takauchi (2015).

4

each firm can diﬀer in several ways (e.g., in quality) since final goods are diﬀerentiated.7 While
the firms freely supply their products domestically, they incur a per unit transport cost of τ ≥ 0
to export.
The representative consumer’s utility function in country i is
1
2
Ui = yi + qii + qji − (qii2 + qji+ 2bqii qji ), i ̸= j; i, j = H, F,
2where yi is the numeraire good. This utility function yields the following inverse demands:
pii = 1 − qii − bqji

(1)

pij = 1 − qij − bqjj .
In equation (1), pii (pij ) denotes firm i’s product price in country i (j), qii (qjj ) denotes firm i’s(j’s) domestic supply, and qij (qji ) denotes firm i’s (j’s) exports to country j (i). The parameterb ∈ [0, 1] measures the degree of product substitutability between the final goods of firms H
and F . If b is zero, then the firms become monopolists; whereas if b = 1, firm i’s productbecomes perfectly substitutable by firm j’s product. We assume that b is determined by firm
i’s investment in product R&D, di ∈ {0, 1/2}, as follows: b ≡ 1 − (dH + dF ).8
We consider a binary-choice case in which firms choose whether to invest in R&D by expending a fixed cost or not. It is well-known that fixed costs are needed in order to conduct
R&D (Desmet and Parente, 2010; Dubois et al., 2015; Tang, 2006; Aboal and Garda, 2016).
Hence, firms often opt for “do not invest in R&D” due to such fixed costs. For example, inthe manufacturing and service industries in Germany, there is a number of firms which R&D
expenditures are close to zero (Czarnitzki and Hottenrott, 2011; Peters, 2009).9 Despite theseempirical facts, the framework in which firms decide their investment level of R&D is prevalent,
7

Even if these two inputs were perfectly homogeneous, firms use only their domestic input when there is a
suﬃciently high tariﬀ on imported inputs.
8This formulation of b is the same type employed by Lin and Saggi (2002). However, they assume thatfirm’s investment in product R&D di is a continuous variable, and the cost function of product R&D is convex
with respect to di . Moreover, although Lin and Saggi (2002) further add to their model a stage-game in whichfirms conduct cost-reducing R&D investment as in d’Aspremont and Jacquemin (1988), from the purpose of ouranalysis, we do not consider cost-reducing R&D investment.
9
In the United Kingdom and Germany, Bond et al. (2005) similarly found that there are some firms that do
not conduct R&D in the high tech sector.

5leading existing studies to exclusively focus on positive investment levels (i.e., an interior equilibrium), and the option “do not invest in R&D” is virtually ignored (d’Aspremont and Jacquemin,
1988; Lin and Saggi, 2007; Rosenkranz, 2003; Takauchi, 2015). To explicitly consider the option“do not invest in R&D”, Lambertini and Rossini (1998) build a binary-choice model of productR&D and examine on the firm’s strategic decision whether to invest in R&D or not. Similar totheir analysis, we focus on the actual options of firms, and consider the situation in which firms
strategically choose whether to invest in R&D or not.10
Let I be the strategy “invest in R&D” and N the strategy “do not invest in R&D”. If firm
i chooses I, then di = 1/2 and it pays the fixed investment cost k > 0. If it chooses N , thendi = 0 and it has no investment cost. Hence, when all firms invest, b = 0; while b = 1 when
no firm invests. If only one firm invests, then b = 1/2. In Section 4, we relax the assumption
on di and consider a general case where di = d ∈ (0, 1/2] if firm i invests. Under this extended
setting, our main results basically hold.
Firm i’s gross profit (excluding k) is Πi ≡ (pii − wi )qii + (pij − wi − τ )qij , where wi is theprice of the input produced by supplier i. Supplier i makes a take-it-or-leave-it oﬀer, and its
profit is πi ≡ (wi − w)(qii + qij ), where w is the unit cost. For simplicity, we set w equal tozero.11
We consider the following three-stage game: In the first stage, each firm independently andsimultaneously chooses whether to invest in product R&D (I) or not (N ). In the second stage,
each supplier decides its input price. In the third stage, firms compete à la Cournot in the H
and F markets. R&D activity takes a significant amount of time; firms generally implement
R&D activity as a long-term project and it is impossible to frequently change decisions withina short period of time. Thus, firms make decisions on R&D investment in the first stage of thegame. In contrast, the production quantity decision is easier to change, and firms, therefore,
10

Matsushima and Mizuno (2009), Mukherjee and Mukherjee (2013), Poddar and Bibhas (2010), and Zanchettin
and Mukherjee (2017) also adopt a binary-choice of product R&D.
11
This setting does not alter our results.

6

determine their outputs in the last stage.
Since firms have two options, four cases can arise: II, IN , N I and N N . All firms invest in
II. In IN (N I), firm H chooses I (N ) and firm F chooses N (I). No firm invests in N N . The
solution concept is the subgame perfect Nash equilibrium (SPNE).

3

Results

We solve the game by backward induction, so we first consider the third stage of the game.
3rd stage: The first-order conditions to maximize firm i’s profit are∂Πi= 0 ⇔ 1 − bqji − 2qii − wi = 0,∂qii
∂Πi
= 0 ⇔ 1 − 2qij − bqjj − wi − τ = 0 for i ̸= j.
∂qij
These yield the third-stage outputs:qii (wi , wj ) =2 − b + bwj − 2(τ + wi )2 − b + b(τ + wj ) − 2wi; qij (wi , wj ) =.
4 − b24 − b2

2nd stage: Using the third-stage outputs, the profit maximization problem maxwi π(wi , wj ),yields the following best response function of supplier i:
wi = BRi (wj , b) ≡1((2 − b)(2 − τ ) + bwj ) for i ̸= j.
8Hence, the input price shows strategic complementarity. We can explain this result as follows.
Suppose that supplier j raises its price; then, a rise in wj increases firm j’s production cost andreduces outputs. A reduction in the rival’s outputs tends to increase firm i’s outputs, whichthen increases the input demand for firm i, influencing supplier i to raise its price. Therefore,
there is a strategic complementarity between input prices.
From BRi (wj , b), the second-stage input price becomeswi (b) =(2 − b)(2 − τ ).
2(4 − b)7

1st stage: Plugging the second-stage outcomes into the profit of firm i, we haveΠi (b) =16(2 − b)2 (1 − τ ) + (b4 − 4b3 − 8b2 + 16b + 80)τ 2
2(4 − b)2 (4 − b2 )2.

(2)

Substituting b = 1 − (dH + dF ) in (2) and using the four possible combinations of dH and dF ,(dH , dF ) = {(0, 0), (0, 1/2), (1/2, 0), (1/2, 1/2)}, we obtain firm i’s equilibrium profit:5τ 2 − 4τ + 4
85τ 2 − 16τ + 16
; ΠII
=
,
i
162
32
2
I = ΠIN = 2(1369τ − 576τ + 576) ,
ΠN
i
i
11025

N =
ΠN
i

(3)

where the superscripts on the variables denote the equilibrium regimes.
The equilibrium input prices are
wiN N =2−τ3(2 − τ )2−τ; wiII =; wiN I = wiIN =
.
6
4
14

(4)

Firm i’s equilibrium outputs are
4 + 7τ N N
4 − 11τ
2 + τ II2 − 3τ
; qij =
; qiiII =; qij =,
1818
8
8
24 + 23τ N I
IN = 24 − 47τ .
qiiN I = qiiIN =; qij = qij
105
105

qiiN N =(5)

To ensure a positive quantity in all regimes, we assume the following.
Assumption 1. τ <4.
11Equations (3) and (4) yield the following lemmas.
Lemma 1. (i) Suppose that firm j chooses N . Then, if ϕl (τ ) > k, firm i (i ̸= j) chooses I;
otherwise, it chooses N . (ii) Suppose that firm j chooses I. Then, if ϕu (τ ) ≤ k, firm i (i ̸= j)chooses N ; otherwise, it chooses I. Here,NN
ϕl (τ ) ≡ ΠIN=i −Πi1136−1136τ −54841τ 27236−7236τ −32491τ 2
NI; ϕu (τ ) ≡ ΠII
=
.
i −Πi
198450352800

N N > (≤) 0 ⇔ ΠN I − k − ΠN N > (≤) 0 ⇔ ϕ (τ ) > (≤) k andProof. From (3), ΠINl
H − k − ΠH
F
F
NI
II
IN
ΠII
H − k − ΠH ≤ (>) 0 ⇔ ΠF − k − ΠF ≤ (>) 0 ⇔ ϕu (τ ) ≤ (>) k. □Lemma 2. (i) wiII > wiN I = wiIN > wiN N . (ii)8∂wir∂τ < 0, where r = II, N I, IN, N N .

Proof. (i) From (4), wiII − wiN I = (2 − τ )/28 > 0 and wiN I − wiN N = (2 − τ )/21 > 0. (ii)∂wiN N /∂τ = −1/6, ∂wiII /∂τ = −1/4 and ∂wiN I /∂τ = −3/14. □Lemma 1 yields Proposition 1.
√

355−142). (i) If k < ϕl (τ ), then II occurs. (ii)
Proposition 1. 1. Suppose that τ < τ ≡ 4(105 54841

If ϕl (τ ) ≤ k ≤ ϕu (τ ), then N N and II can occur. (iii) If k > ϕu (τ ), then N N occurs.
2. Suppose that τ ≥ τ . (i) If k < ϕu (τ ), then N N and II can occur. (ii) If k ≥ ϕu (τ ), thenN N occurs.
[ 4])+585037τ 2, ϕu (τ ) − ϕl (τ ) = 46948(1−τ> 0,Proof. From Lemma 1, ϕu (τ ) > 0 ∀τ ∈ 0, 113175200568∂ϕl (τ )/∂τ = − 568+54851τ
< 0 and ϕl (0) = 99225≡ k. Solving ϕl (τ ) = 0 with respect to τ , we99225√

355−142)have τ ≡ 4(105 54841≃ 0.13394. □

0.025

k

NN

ϕu (τ)

k

0.015

NN
II

k
ϕl (τ)
0.005

τ

II
0
0

0.1

0.2

0.3

4
11

τ

Figure 1: SPNE of the game in (τ, k)-space

We first consider Lemma 2. Part (i) states that input prices increase with the number ofinvesting firms. This result depends on the opportunistic behavior by the input suppliers. Eachfirm invests in R&D in the first stage of the game, and each input supplier charges its price in thesecond stage. Then, if supplier i charges a higher input price after observing firm i’s investment,
then the supplier can extract the R&D benefit of firm i.12 Therefore, when firm i invests in
12

Prior studies show that opportunistic behaviour by upstream agents may cause a hold-up problem in down-9

R&D, the input price wi becomes higher. In addition, as we find in the best response function
BRi (wj , b), there is a strategic complementarity between home and foreign inputs. Hence, if
firm F invests in the case in which firm H invests, the input prices in both H and F are higher
than those when only one of the firms invests.
Part (ii) is intuitive. A rise in the transport cost τ continuously reduces input prices because
a rise in τ hinders exports and discourages production, thereby reducing the input demand.
Therefore, the supplier tries to restore input demand by lowering its price.
As we show in Proposition 1, only “N N ”, “II” or “N N and II” can become equilibriumoutcomes of the game. Hereafter, we call N N the “no one invests” regime and II the “everyone
invests” regime.
The intuition behind Proposition 1 is as follows. The R&D motive depends on the transport
and investment costs. When investment cost k decreases, the R&D motive intensifies because
the net benefit of product diﬀerentiation increases. Furthermore, firms gain the diﬀerentiation
benefit from both domestic and foreign markets. When τ falls, the R&D motive intensifies
because a lower τ raises exports and increases the diﬀerentiation benefit gained from the foreign
market. Hence, if τ and k are small enough, I becomes the dominant strategy and the “everyone
invests” regime appears. If both costs are large enough, N becomes the dominant strategy andthe “no one invests” regime appears (see Fig. 1).
When τ is high and k is at an intermediate level, both “no one invests” and “everyone invests”
regimes can appear. If firm i deviates from the “everyone invests” regime, wi falls. However, wj
also falls due to strategic complementarity. The deviation makes competition tougher and therival’s cost does not increase; hence, firm i does not deviate from the “everyone invests” regime.
Additionally, the deviation from the “no one invests” regime raises the input price, but it brings
a diﬀerentiation benefit to the deviator. On the other hand, in this case, τ is high and the size
stream investment. Banerjee and Lin (2003) present a long-term input price agreement (i.e., a fixed-price contract)
to resolve this hold-up problem. In contrast, Takauchi and Mizuno (2019) show that solving the hold-up problem
with a long-term input price agreement can harm all firms.

10

of the foreign market is small, and hence the diﬀerentiation benefit from the foreign market issmall. Since the R&D benefit is relatively small, there is no incentive to deviate from the “noone invests” regime.
Next, we examine the eﬀects of the transportation cost on consumers. From the utility
function, the formula for consumer surplus isCSi = Ui − (yi + pii qii + pji qji )12= (qii2 + qji+ 2bqii qji ).
2
The above CSi , (1) and (5) yield
CSiN N =2(2 − τ )2
4 − 4τ + 5τ 2
; CSiII =
.
81
64

(6)

From (6), we establish the following.
Proposition 2. (Consumer-benefiting transport cost)](9329. If N N appears within the area in which the multiple equilibria of
1. Suppose k ∈ 0, 10672200“N N and II” arise in Figure 1, then τ = ϕ−1
l (k) maximizes the consumer surplus.
2. Suppose k ∈

(

)

9329
10672200 , k . (i) If N N appears within the area in which the multiple equilibria

of “N N and II” arise in Figure 1, then τ = ϕ−1
l (k) maximizes the consumer surplus; (ii) if II
appears in that area, then τ = ϕ−1
u (k) maximizes the consumer surplus.
[)
3. Suppose k ∈ k, k . If II appears within the area in which the multiple equilibria of “N N
and II” arise in Figure 1, then τ = ϕ−1
u (k) maximizes the consumer surplus.
568
201
Here, k ≡ 99225≃ 0.00572436 and k ≡ 9800
≃ 0.0205102.

201= ϕu (0), whereas
Proof. First, from Proposition 1 and its proof, N N &II appear if k < k ≡ 9800

( )only N N appears if k ≥ k. In the interval 0, k , the equilibrium transition is divided into the(]( 9329)9329
following three types: for k ∈ 0, 10672200, N N &II → II as τ decreases; for k ∈ 10672200,k ,[)N N → N N &II → II as τ decreases; and for k ∈ k, k , N N → N N &II as τ decreases. Here,

11

4
9329
201
ϕu ( 11) = 10672200≃ 0.00087414 and k ≡ ϕu (0) = 9800
≃ 0.0205102.
)
Second, ∂CSiN N /∂τ = − 4(2−τ< 0 and ∂CSiII /∂τ = − 2−5τ81
32 < 0, so the consumer sur-plus in both aforementioned regimes is monotonically decreasing for τ . Simple algebra yields
18, min CSiN N = CSiN N τ =4/11 = 121
and min CSiN N − max CSiII =max CSiII = CSiII τ =0 = 1671936 > 0. These imply Proposition 2. □

0.09

CSiNN

0.08

0.07

0.06

CSiII
NN
II

0.05

II
0

0.057006

NN
0.3202074τ
11

Figure 2: The Relationship between CSi and τ : k = 0.0045.

The input price discontinuously rises as the number of investing firms increases (part (i)of Lemma 2). Hence, the input price in the “everyone invests” regime is highest among all
regimes. The firms’ production cost also sharply rises due to the sharp increase in input prices.
Because this rapid increase in the cost suppresses the production of final good, the total output
plummets. Consequently, for consumer surplus, min CSiN N > max CSiII holds. (See proof of
568
Proposition 2.) For example, when k < k (= 99225), the equilibrium transition N N &II→IIarises as τ decreases (see Fig. 1). In any regime, a decrease in τ raises consumer surplus; hence,
the threshold at which the equilibrium switches from N N &II to II maximizes consumer surplus
if N N appears in N N &II (see Fig. 2).
Proposition 2 has a significant policy implication associated with competition (antitrust)policy. In the United States, for example, the consumer welfare standard is currently used, andthis position can be summarized by the statement: “If consumers are made worse oﬀ then the12practice is to be prohibited. Otherwise, it is to be allowed.” (Viscusi et al., 2018, p. 97.) A
reduction in the transport cost usually lowers the entry barrier for foreign competitors, facilitates
imports, promotes competition in the domestic market, and decreases the price, so one could
think that transport cost reduction makes consumers better oﬀ. Therefore, when the consumer
welfare standard is employed, it is naturally assumed that the authority in charge of competition
policy has an incentive to support a transport cost reduction and promote competition in orderto protect consumers. However, Proposition 2 indicates that there is a case in which it is notdesirable to use the consumer welfare standard, and that such case is not necessarily rare. In
this regard, our result provides a new insight into the context of competition (antitrust) policy
and international trade.
While Proposition 2 shows a result that contradicts with the common wisdom in trade theory, it is broadly consistent with empirical findings. Some empirical studies (Caves et al. 1991,
Grabowski and Vernon 1992, and Thomadsen 2007) have already found several examples inwhich the promotion of competition leading to new entries of firms into the market, and theresulting increase in the number of operating firms, ends up raising prices, potentially harming
consumers.13 Generally, the transport cost is equivalent to an entry barrier that shuts foreignexporting firms out if the level is high enough. Hence, a transport cost reduction lowers thebarrier hindering supplying activities by foreign firms and promotes trade, so a transport cost
reduction intensifies the domestic market competition. Proposition 2 states that the promotion of competition through a transport cost reduction can raise prices and harm consumers.
Therefore, it constitutes a theoretical evidence that supports empirical findings showing thatthe enhancement of competition can raise prices.

13

Caves et al. (1991) empirically show that market entry can lead to an increase in drug prices. Grabowski
and Vernon (1992) also shows that an entry causes a rise in the price of pharmaceuticals. Using a simulation,Thomadsen (2007) demonstrates that duopoly price can be higher than monopoly prices in a first-food market.

13Social welfare
Unlike consumer surplus, social surplus tends to increase with transport cost reduction. We
verify this fact here.
Excluding the investment cost k, the social surplus in country i, SWi , is the sum of theconsumer surplus, CSi , the profit of firm i, Πi , and the profit of the input supplier i, πi . The
equilibrium outcome becomes
SWiN N =56 − 56τ + 95τ 2;
162

SWiII =

28 − 28τ + 19τ 2.
64

As we show in Proposition 2, the consumer surplus is largest in the “no one invests” regime.
However, the profit ranking of firm i is ambiguous because in the “everyone invests” regime,
the firm’s profit rises as the transport cost falls, while in the “no one invests” regime, its profit
can decrease as the transport cost falls. In the “everyone invests” regime, each firm becomes amonopolist in both the home and foreign markets, so the firm’s profit increases as the transport
cost falls. In contrast, in the “no one invests” regime, the firms’ products are homogenous, andeach firm faces tough competition. Because high transport cost shuts out the rival’s exports,
each firm can enjoy a domestic monopoly. A prohibitive transport cost maximizes each firm’s
profit.14
Since the input price jumps with the number of investing firms (Lemma 2), the profit ofsupplier i, πi , is the largest in the “everyone invests” regime. In our model, the input supplier’s
profit strongly aﬀects the welfare ranking. Comparing the welfare levels between II and N N ,we have((SWiII − k) − SWiN N =)476 − 476τ − 1501τ 2
−k.
5184
|
{z
}
>0

As long as k is small enough, SWiII > SWiN N holds.

14
This property of a firm’s profit is the same as in Brander and Krugman (1983). For a more detailed argument,
see Takauchi and Mizuno (2019).

14

In the “everyone invests” regime, consumer surplus, firm’s profit and the input supplier’s
profit are monotonically decreasing with respect to the transport cost, while in the “no one
invests” regime, a change in the firm’s profit can be the opposite of the other items.15 For thisreason, in the “no one invests” regime, the social surplus is U-shaped for the transport cost.16

0.45

SWiII

0.4

0.35

SWiNN
0.3

NN
II

NN

II

τ

0

0.057006

0.320207

4
11

Figure 3: The Relationship between SWi and τ : k = 0.0045.

Figure 3 illustrates the relationship between social surplus and transport cost. If the investment cost is small enough, that is, k < k, and the “everyone invests” regime appears, then thewelfare level may jump upwards due to the reduction in transport cost.17
As we show in our arguments so far, the government has an incentive to activate firms’
innovation by reducing trade barriers, while consumers do not necessarily have such an incentive.
There is a market structure in which consumers prefer a certain level of trade barriers, so our
result points out that the government needs to be careful when promoting trade liberalization.

In fact, ∂SWiII /∂τ = (19τ − 14)/32 < 0. On the other hand, ∂SWiN N /∂τ = (95τ − 28)/81; ∂SWiN N /∂τ > 0
for τ > 28/95 (< 4/11).
16
SWiN N is U-shaped with respect to the transport cost, and it has the same property as in Brander andKrugman (1983). Helpman and Krugman (1985) also illustrate a U-shaped welfare curve in Figure 5.11, page 110of their work.
17
Even if we consider k ∈ (k, k), a similar result holds.
15

15

4

Extension

The degree of product diﬀerentiation
In the previous section, we assumed that di = 1/2 if firm i invests in product R&D. Here, we
relax this assumption and consider a more general case in which di = d ∈ (0, 1/2] if firm iinvests, and di = 0 otherwise. The other settings are the same as in the previous section. We
will see that even if we consider that di is within the range (0, 1/2], our main previous results
basically hold.
We first examine the equilibrium of the game. In the first stage of the game, each firm
decides whether to invest (I) or not (N ). Substituting di = d into (2), we obtain the gross profit
of firm i in the “everyone invests” regime:ΠIIi (d) =

(16d4 − 56d2 + 16d + 85)τ 2 − 16(2d + 1)2 τ + 16(2d + 1)2.
2(3 + 2d)2 (3 + 4d − 4d2 )2

By taking di = d and dj = 0, the gross profits of firm i in IN and N I becomeIINΠNi (d) = Πi (d) =(d4 − 14d2 + 8d + 85)τ 2 − 16(d + 1)2 τ + 16(d + 1)2.
2(9 + 9d − d2 − d3 )2

We obtain the profit of firm i in the “no one invests” regime by substituting di = 0 into (2), justN in equation (3).
as we did with ΠNi

By using the above equations for the firms’ profit, we now investigate the incentives in theNN .
investment decision. When the rival firm does not invest, firm i invests if ΠINi (d) − k ≥ Πi
N N . Similarly,
Then, the minimum value of k that yields investment is k = Φl (τ, d) ≡ ΠINi (d)−ΠiNIwhen the rival firm invests, firm i also has an incentive to invest if ΠIIi (d) − k ≥ Πi (d).

Solving this inequality for k, we have the minimum value of k with investment: k = Φu (τ, d) ≡NIΠIIi (d)−Πi (d). Comparing these thresholds, we obtain Φu (τ, d) ≥ Φl (τ, d). Then, we establish

Proposition 3.
Proposition 3. 1. Suppose that τ < τ (d). (i) If k < Φl (τ ), then II appears. (ii) If Φl (τ ) ≤16

k ≤ Φu (τ ), then N N and II can appear. (iii) If k > Φu (τ ), then N N appears. 2. Suppose thatτ ≥ τ (d). (i) If k < Φu (τ ), then N N and II can appear. (ii) If k ≥ max{Φu (τ ), 0}, then N N
appears.

Proof. See Appendix A.

Next, we consider the relationship between consumer surplus and transport cost. Because
we added the new parameter d into the model, our analysis is more complicated. Thus, toprovide a simple confirmation on whether a result similar to that of Proposition 2 holds, we here
assume that the investment cost k takes a discrete value, k ∈ {0.001, 0.002, 0.003, . . . }. From
Proposition 3, we find that only the “no one invests” regime, “everyone invests” regime and the“no one invests” and “everyone invests” regimes can appear in equilibrium. Hence, we compare
the consumer surplus of the two regimes. Country i’s consumer surplus in the “everyone invests”
regime is
CSiII (d) =16d5 τ 2 − 8d3 (11τ 2 − 8τ + 8) + d(93τ 2 − 48τ + 48) + 4(τ − 2)2.
2(2d + 3)2 (3 + 4d − 4d2 )2CSiN N is given by equation (6).
Since Φu (τ, d) is a decreasing function of τ , Φu (τ, d) takes the maximum value at τ = 0.
Moreover, with τ = 0, Φu (0, d) is maximized at d = 1/2. Since Φu (0, 1/2) = 201/9800 ≃ 0.205,
from Proposition 3, for any k > 0.205, no firm decides to invest. In addition, since CSiN N
decreases with τ , consumer surplus is maximized at τ = 0.
The remaining cases are k ∈ {0.001, 0.002, . . . , 0.204}. We define the inverse functions of
−1
−1
−1
NN
Φ−1
u (τ, d) and Φl (τ, d) by Φu (k, d) and Φl (k, d), respectively. Numerically comparing CSi

with CSiII (d), we establish the following result.
Proposition 4. Let k ∈ {0.001, 0.002, · · · }. 1. Suppose that N N appears in the N N and
II regimes. (i) For k ≤ Φl (0, d), consumer surplus is maximized at τ = Φ−1l (k, d). (ii) For
k > Φl (0, d), consumer surplus is maximized at τ = 0.
17

2. Suppose that II appears in the N N and II regimes. (i) For k ≤ Φu (4/11, d), consumer surplus
is maximized at τ = 0 if d > 0.479; otherwise, it is maximized at τ = Φ−1u (k, d) if d ≤ 0.479.
(ii) For Φu (4/11, d) < k ≤ Φu (0, d), consumer surplus is maximized at τ = Φ−1u (k, d). (iii) For
k > Φu (0, d), consumer surplus is maximized at τ = 0.
Proof. See Appendix B.

5Conclusion

Reducing transport cost in two-way trade increases each country’s total outputs and lowers
prices due to the promotion of competition in the domestic market, so a reduction in transport
cost will certainly make consumers better oﬀ. Thus, a zero transportation cost, that is, free
trade, maximizes consumer surplus. Although we could consider this assertion as theoreticallyobvious, it is not necessarily true. We show that when an exporting firm that deals exclusively
with its domestic input supplier engages in product R&D, a positive transport cost can maximize
consumer surplus. We find this result in a setting where R&D race and input price are present.
When the investment cost is not too high, the equilibrium pattern also depends on the transportcost. Whilst the “everyone invests” regime appears if both the investment and transport costs
are low, the “no one invests” regime appears if both costs are high. When both costs are at anintermediate level, then, multiple equilibria, the “everyone invests” and “no one invests” regimes,
appear. The input suppliers can extract R&D benefits by charging a higher price after observing
their customers’ investments, so input prices increase when firms invest. In addition, there isa strategic complementarity between the domestic and foreign input prices; hence, the input
price increases with the number of investing firms. Therefore, the “everyone invests” regime
yields the worst production eﬃciency of all regimes and the smallest consumer surplus througha decline in total outputs. In the case of low investment cost, if the transport cost declines below
a certain level, the consumer surplus decreases because the equilibrium can switch from the “no18one invests” regime to the “everyone invests” regime.
Our model is based on a product R&D model in which investment promotes a degree ofhorizontal product diﬀerentiation (Lin and Saggi, 2002; Lambertini and Rossini, 1998), so we
do not examine the vertical product diﬀerentiation case. Also, considering quality-improving
R&D, for example, may be fruitful, but it is beyond the scope of this study and is left to future
research.

Acknowledgements
This paper is a substantially revised version of “Takauchi and Mizuno (2019), Discussion Paper
No. 1911 (November 6, 2019), Graduate School of Economics, Kobe University.” We speciallythank Noriaki Matsushima, Akio Kawasaki, and Eiichi Miyagawa for their useful comments. In
particular, Eiichi Miyagawa and Noriaki Matsushima have provided us with valuable suggestions.
We also thank seminar participants at the 2020 JEA spring meeting for comments. Takauchi
acknowledges financial support from the Japan Society for the Promotion of Science (JSPS),
KAKENHI Grant Number 20K01646. All errors are our own.

Appendix A. Proof of Proposition 3
To prove this proposition, we show that (i) Φl (τ, d) < Φu (τ, d), (ii) Φl (0, d) > 0 and Φu (0, d) > 0,(iii) ∂Φl /∂τ < 0 and ∂Φu /∂τ < 0, (iv) Φl (0, d) > Φu (4/11, d), (v) τ (d) ∈ [0, 4/11] such thatΦl (τ (d), d) = 0 and (vi) Φu (4/11, d) > 0 for d > 0.479 and Φu (4/11, d) ≤ 0 for d ≤ 0.479. Fromthese conditions, we can create a similar figure as in Figure 1, so it is enough to show them tocomplete the proof.
First, we show that Φl (τ, d) < Φu (τ, d). The diﬀerence between the two isΦu (τ, d) − Φl (τ, d) =d2 (ψ2 τ 2 + ψ1 τ + ψ0 ),81(3 + 2d)2 (3 + 4d − 4d2 )2 (9 + 9d − d2 − d3 )2

where ψ2 ≡ 2720d10 + 8160d9 − 56896d8 − 183408d7 + 337514d6 + 1298646d5 − 504621d4 −19

3471012d3 − 632448d2 + 3223638d + 1600155, ψ1 ≡ −16(2d2 + 3d + 1)2 (8d6 − 180d4 + 81d2 + 1458)
and ψ0 ≡ 16(2d2 + 3d + 1)2 (8d6 − 180d4 + 81d2 + 1458). Thus, the sign of Φu (τ, d) − Φl (τ, d)depends on the numerator. The numerator is a quadratic function of τ and the coeﬃcient of τ 2
is positive. Further, its discriminant is ψ12 − 4ψ2 ψ0 = −5184d4 (2d2 + 3d + 1)2 (2d2 + 6d + 3)(4d4 −45d2 + 81)2 (8d6 − 180d4 + 81d2 + 1458) < 0, and hence Φl (τ, d) < Φu (τ, d).
Second, we consider the signs of Φl (0, d) and Φu (0, d). Substituting τ = 0 into Φl (τ, d) andΦu (τ, d), we haveΦl (0, d) =24d2 (18 − 5d2 )
8d2 (18 − d2 )> 0 and Φu (0, d) =
> 0.
2
4
2
(81 − 45d + 4d )81(9 − d2 )2Third, we consider the sign of the first derivatives of Φl (τ, d) and Φu (τ, d) with respect to τ .
We haved(ψ1′ τ + ψ0′ )∂Φl (τ, d)=,∂τ
(2d + 3)2 (3 + 4d − 4d2 )2 (d3 + d2 − 9d − 9)2
where ψ1′ ≡ −48d9 − 32d8 + 840d7 − 7935d5 − 3114d4 + 24618d3 + 15876d2 − 19467d − 13122 < 0and ψ0′ ≡ −24d(2d2 + 3d + 1)2 (18 − 5d2 ) < 0. Note that we confirm the first inequality by using
a numerical calculation. Since ψ1′ < 0 and ψ0′ < 0, the above derivative takes a negative value:∂Φl (τ, d)/∂τ < 0.
We consider the sign of ∂Φu (τ, d)/∂τ .
∂Φu (τ, d)ψ1′′ τ + ψ0′′=,∂τ
81(9 + 9d − d2 − d3 )2
where ψ1′′ ≡ −85d5 −170d4 +1526d3 +3060d2 −6489d−13122 < 0 and ψ0′′ ≡ −8d(1+d)2 (18−d2 ) <0. Note that we find the first inequality by using a numerical calculation. Hence, we have∂Φu (τ, d)/∂τ < 0.

20

Fourth, we show that Φl (0, d) > Φu (4/11, d).
Φl (0, d) − Φu (4/11, d) =



11
10
9
8
 −7744d − 23232d + 152960d + 513696d






7
6
5
4
8d  +204056d − 615384d − 905031d − 2110698d 





−4152060d3 − 2248236d2 + 1416447d + 1062882

9801(3 − 2d)2 (3 − d)2 (d + 1)2 (d + 3)2 (2d + 1)2 (2d + 3)2> 0,

To show the above inequality, we use a numerical calculation.
Fifth, we consider τ (d) such that Φl (τ (d), d) = 0. Solving Φl (τ, d) = 0 for τ , we have[]√
4 2d5 + 4d4 − 34d3 − 72d2 − 36d − 9(d − 3)(d + 1)(d + 3) d(d + 2)(18 − d2 )
τ = τ (d) ≡.
85d5 + 170d4 − 1526d3 − 3060d2 + 6489d + 13122√
From numerical calculation, we obtain 0 ≤ τ (d) < 40(3219 14053 − 21620)/134273093 ≃ 0.107.
Finally, we consider the sign of Φu (4/11, d). We have 48d + 32d + 3780d + 13860d + 6318d − 39852d 

8d 


−77517d3 − 40824d2 + 15309d + 13122
9Φu (4/11, d) = −



8

7

6

5

4

121(2d + 3)2 (−4d2 + 4d + 3)2 (−d3 − d2 + 9d + 9)2

The sign of Φu (4/11, d) depends only on the terms in the square brackets of the numerator. We
solve the equation numerically and find that Φu (4/11, d) > 0 if and only if d > 0.479. Therefore,
we complete the proof of Proposition 3. □Appendix B. Proof of Proposition 4
We show that (i) CSiN N > CSiII (d), (ii) CSiN N and CSiII (d) decrease with τ and (iii) at τ = 0,
CSiN N and CSiII (d) take positive values.
First, we consider CSiN N − CSiII (d).


 16(16d − 72d + 81)(2d + 1) − 16(16d − 72d + 81)(2d + 1) τ 

d


5
4
3
2
2
+ (256d − 1040d − 1088d + 5976d + 1008d − 6237)τ
3

CSiN N −CSiII (d) =2

3

162(3 − 2d)2 (1 + 2d)2 (3 + 2d)22.

The sign of CSiN N −CSiII (d) depends only on the terms in the square brackets. Using numerical21

calculation, we find that the coeﬃcient of τ 2 in the square brackets takes a negative value andthat two roots for τ satisfying CSiN N − CSiII (d) = 0 are not in [0, 4/11). Hence, for anyτ ∈ [0, 4/11), CSiN N − CSiII (d) > 0 holds.
Second, we show that CSiN N and CSiII (d) decrease with τ . The first derivative of CSiN N
with respect to τ is ∂CSiN N /∂τ = −4(2 − τ )/81 < 0. The derivative of CSiII (d) is∂CSiII (d)(16d5 − 88d3 + 93d + 4)τ − 8(1 − d)(2d + 1)2=.
∂τ(3 + 2d)2 (3 + 4d − 4d2 )2The numerator determines the sign of the above equation. Since the coeﬃcient of τ is positiveand τ ∈ [0, 4/11), the numerator takes the supremum at τ = 4/11. Substituting τ = 4/11 into
the numerator, we have 4(−18 + 27d + 16d2 )/11, which takes a negative value over d ∈ [0, 1/2].
Hence, ∂CSiII (d)/∂τ < 0.
Finally, at τ = 0, we have CSiN N = 8/81 > 0 and CSiII (d) = 8(1 − d)/(9 − 4d2 )2 > 0.
Now, we are ready to prove Proposition 4. First, we consider a case where N N appears in
the N N and II regimes. Because ∂CSiN N /∂τ < 0 and ∂CSiII (d)/∂τ < 0, the candidates that−1
maximize consumer surplus are τ = 0 and τ = Φ−1l (k, d), where Φl (k, d) is an inverse functionof k = Φl (τ, d).
If k ≥ Φl (0, d), then for any τ , the N N regime appears. Hence, τ = 0 maximizes consumer
surplus. On the other hand, if k < Φl (0, d), we need to compare the consumer surplus at τ = 0
and τ = Φ−1l (k, d).
Since Φl (τ, d) decreases with τ , Φl (τ, d) is maximized at τ = 0. In addition, since Φl (0, d)
increases with d, the maximum value of Φl (τ, d) is Φl (0, 1/2) = 568/99225 ≃ 0.0057. Hence, forany k ∈ {0.006, 0.007, . . . }, N N is the only regime, which means that τ = 0 maximizes consumer
surplus.
The remaining cases are in k ∈ {0.001, . . . , 0.005}. Because ∂Φl (0, d)/∂d = 32d/(9−d2 )3 > 0,
there exists a d′ such that for any d ∈ [0, d′ ), we have k > Φl (0, d). This case also leads to the
maximum consumer surplus with τ = 0. Since we assume that k takes a discrete value, we can22numerically solve k = Φl (0, d′ ) for d′ by fixing k to a certain value.
Over d ∈ [d′ , 1/2], there always exists τ = Φ−1l (k, d), which is the candidate that maximizes
consumer surplus. Fixing k to a certain value, k ∈ {0.001, . . . , 0.005}, and substituting τ = 0
II
N N , respectively, we can compare the consumer surplus
and τ = Φ−1l (k, d) into CSi (d) and CSi

in these cases. Since CSiII (d) and CSiN N depend only on d, we can compare them numericallyand find that for any k ∈ {0.001, . . . , 0.005}, CSiN N > CSiII (d). Hence, τ = Φ−1l (k, d) maximizes
consumer surplus. Summarizing the discussion above, we obtain the first part of Proposition 4.
Next, we show the second part. We assume that II occurs in the N N and II regime.
Since for some d, Φu (τ, d) takes a positive value at τ = 4/11, we calculate this threshold value.
Numerically solving Φu (4/11, d) > 0 for d, we obtain d > 0.4879.
To show the second part, we consider two cases: (i) k < Φu (4/11, d) and (ii) k ≥ Φu (4/11, d).
Note that we can consider case (i) only for d > 0.4879. In case (i), only the II regime appears.
Hence, τ = 0 maximizes consumer surplus because CSiII (d) decreases with τ .
We now consider case (ii). Since Φu (τ, d) decreases with τ and Φu (0, d) increases with d, themaximum value is Φu (0, 1/2) = 201/9800 ≃ 0.0205. Hence, if k ∈ {0.021, 0.022, . . . }, for any τ ,the II regime occurs, which means that τ = 0 maximizes consumer surplus.
Next, we consider the case with k ∈ {0.001, . . . , 0.020}. Here, we have two candidates thatmaximize consumer surplus: τ = 0 and τ = Φ−1u (k, d). Given some value of k and substitutingII
N N , respectively, we can compare the consumer
τ = 0 and τ = Φ−1u (k, d) into CSi (d) and CSi

surplus in these cases. Since CSiII (d) and CSiN N depend only on d, we can compare them
numerically and find that for any k ∈ {0.001, . . . , 0.020}, CSiN N > CSiII (d). Therefore, we
obtain the second part of Proposition 4. Combining these results, we complete the proof. □23

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