Consumer-benefiting transport cost: The role of productinnovation in a vertical structure
Kazuhiro Takauchi
Tomomichi Mizuno
November 6,2019
Discussion Paper 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†
November 6, 2019
Abstract
Contrary to the standard belief, we show that a positive transport cost can maximize consumer’s
surplus if exporting firms engage in product R&D and use their domestic inputs.

Key words: Transport cost; Consumer’s 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; Fax: +816-6339-7704.
†Graduate School of Economics, Kobe University, 2-1 Rokkodai-cho, Nada-ku, Kobe-City, Hyogo 657-8501,
Japan. E-mail: mizuno@econ.kobe-u.ac.jp.

1Introduction

It is well-known that, in oligopolistic intra-industry trade, a transport cost reduction increases
consumer’s surplus. For example, Helpman and Krugman (1985, p. 108) said, “Trade has a
procompetitive eﬀect: each firm’s exports do not displace an equal volume of shipments fromthe other firm to its home market, so total output and consumption rise and the price falls.”
According to such argument, since a decline in transportation cost facilitates exports, it increases
total output, reduces price, and puts consumers in an advantageous position. Contrarily, weshow that a positive transport cost can maximize consumer’s surplus, if final-good exporting
firms engage in product R&D and use their domestic inputs.
Our result occurs for the following reason. When exporting firms invest, their products
become more diﬀerentiated and competition declines; this increases the momentum of finalgood production, thereby increasing the input demand. After observing demand growth, input
suppliers increase prices to extract R&D benefit,1 which raises firms’ production cost. Hence,
the total output and consumer’s surplus decline with investments by all firms. When transport
cost is low, since investment becomes the dominant strategy and all firms invest, the consumer’s
surplus can drop if transport cost decreases.
In the context of oligopolistic intra-industry trade, Bastos and Straume (2012) and Braun(2008) employ Lin and Saggi (2002)-type product R&D model. However, they focus on therole of skilled and unskilled workers and consider the eﬀects of trade liberalization on wage
inequalities and labor demands through the impact on firm’s innovation.
By considering a technology transfer through licensing, Kabiraj and Marjit (2003) show
the existence of a tariﬀ rate that maximizes consumer’s surplus. A high tariﬀ induces foreign
firms’ technology transfer to local firms and increases the total output, thereby increasing theconsumer’s surplus more than that in zero tariﬀ case. However, their study analyzes technology
1

For detailed arguments on this opportunistic behavior by upstream suppliers, see Takauchi and Mizuno (2019).

1

transfer in a one-way trade, and it is quite diﬀerent from our model.
The next section presents the model and results.

2

Model and Results

We consider Brander and Krugman (1983)-type reciprocal market. There are symmetric twocountries, H and F , with final-good market. 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 final-good, firms employ one unit of input. We assume that firm i buys its input froman exclusive supplier i.2 While firms freely supply their products domestically, they incur a perunit transport cost τ ≥ 0 to export.
In country i, firms face the following inverse demands:3
pii = 1 − qii − bqji and pij = 1 − qij − bqjj , i ̸= j; i, j = H, F,(1)where 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 final goods of firms H andF . If b becomes to zero, firms become monopolists; while if b = 1, firm i’s product becomes
perfectly substitutable to firm j’s product. Following Lin and Saggi (2002),4 we assume that b isdetermined by firm i’s investment in product R&D, di ∈ {0, 1/2}, as follows: b = 1 − (dH + dF ).5
In our model, firms have two options such that whether to invest in R&D (labeled I) or not(labeled N ). If firm i chooses I, di = 1/2 and it pays the investment cost k > 0. If it chooses
N , di = 0 and it has no investment cost. Hence, when all firms invest, b = 0; while b = 1, when
2

When there are high trade barriers for importing inputs, firms also use only their domestic inputs.
22
A representative consumer’s utility function is given by Ui = yi + qii + qji − 12 (qii+ qji+ 2bqii qji ), i ̸= j,where yi is the numeraire good. The consumer’s surplus CSi is derived from Ui − (yi + pii qii + pji qji ).
4
See Bastos and Straume (2012), Braun (2008), and Rosenkranz (2003). For a binary-choice case of investment,
see Lambertini and Rossini (1998).
5
Even if we consider the eﬀectiveness of R&D, δ ∈ (0, 1], and assume that b = 1 − δ(dH + dF ), a similar analysis
is possible and our main results do not alter. However, this setting convolutes algebraic analysis. Hence, to avoid
unnecessary complexities, we put δ = 1.
32

no firm invests. If only one firm invests, b = 1/2.
Firm i’s gross profit (excluded k) is Πi ≡ (pii − wi )qii + (pij − wi − τ )qij , where wi is the priceof the input produced by supplier i. Supplier i makes a take-it-or-leave-it oﬀer, and its profit isπi ≡ (wi − w)(qii + qij ), where w is the unit cost. For simplicity, w is set equal to zero.6
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 the level of its input price. In the third stage, firms compete à la Cournotin H and F markets. Since firms have two options, four situations—II, IN , N I, and N N —canarise. All firms invest in II. In IN (N I), firm H chooses I (N ) and firm F chooses N (I). No
one invests in N N . The solution concept is subgame perfect Nash equilibrium (SPNE).
By solving the game using backward induction, we obtain SPNE. The derivation of SPNE
is illustrated in the Technical Appendix. The input prices arewiN N =2−τ
3(2 − τ )2−τ; wiII =; wiN I = wiIN =,6414(2)where the superscript in variables denotes equilibrium regimes.
Firm i’s outputs are
4 + 7τ N N
4 − 11τ
2 + τ II2 − 3τ; qij =; qiiII =; qij =,18188824 + 23τ N I
IN = 24 − 47τ .
qiiN I = qiiIN =; qij = qij105
105

qiiN N =(3)Firm i’s profit areNΠN=i85τ 2 −16τ +165τ 2 −4τ +42(1369τ 2 −576τ +576)
NIIN; ΠII=;
Π=
Π=.
iii
162
3211025(4)

To ensure a positive quantity, we assume τ < 4/11.
(2) and (4) yield 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)
6This setting does not alter our results.

3chooses N . Otherwise, it chooses I. Here,NNϕl (τ ) ≡ ΠIN=i −Πi1136−1136τ −54841τ 27236−7236τ −32491τ 2NI; ϕu (τ ) ≡ ΠII=.
i −Πi198450352800

N N > (≤) 0 ⇔ ΠN I − k − ΠN N > (≤) 0 ⇔ ϕ (τ ) > (≤) k andProof. From (4), ΠINlF
FH − k − ΠHIIINNIΠII
H − k − ΠH ≤ (>) 0 ⇔ ΠF − k − ΠF ≤ (>) 0 ⇔ ϕu (τ ) ≤ (>) k. □Lemma 2. (i) wiII > wiN I = wiIN > wiN N . (ii)∂wir∂τ < 0, where r = II, N I, IN, N N .

N I − w N N = 2−τ > 0. (ii)
Proof. (i) From (2), wiII − wiN I = 2−τ
i28 > 0 and wi21

∂wiN N∂τ= − 16 ,∂wiII∂wiN I13∂τ = − 4 , and ∂τ = − 14 . □Lemma 1 yields Proposition 1.
√355−142)
Proposition 1. 1. Suppose τ < τ ≡ 4(105 54841. (i) If k < ϕl (τ ), II appears. (ii) Ifϕl (τ ) ≤ k ≤ ϕu (τ ), N N and II can appear. (iii) If k > ϕu (τ ), N N appears.
2. Suppose τ ≥ τ . (i) If k < ϕu (τ ), N N and II can appear. (ii) If k ≥ ϕu (τ ), N N appears.
[ 4])+585037τ 2, ϕu (τ ) − ϕl (τ ) = 46948(1−τ> 0,Proof. From Lemma 1, ϕu (τ ) > 0 ∀τ ∈ 0, 113175200∂ϕl (τ )568= − 568+54851τ
< 0, and ϕl (0) = 99225≡ k. Solving ϕl (τ ) = 0 with respect to τ , we have∂τ99225√355−142)
τ ≡ 4(105 54841≃ 0.13394. □

0.025

k

NN

ϕu (τ)

k0.015NNIIk
ϕl (τ)
0.005

τ

II
0
0

0.1

0.2

τ

Figure 1: SPNE of the game

40.3

4
11

The input price jumps up by investment (Lemma 2). A decline in competition due toinvestment facilitates final-good production. However, an increase in the final-good production
expands input demands; hence, each supplier increases its price to raise profit. A rise in thetransport cost τ continuously reduces input prices. This is because a rise in τ impedes exports
and discourages production, thereby reducing the input demand. Therefore, the supplier tries
to restore input demand by lowering its price.
Intuition of Proposition 1 is as follows. The R&D motive depends on transport and investment costs. When the investment cost k decreases, because the net benefit of product diﬀerentiation increases, the R&D motive intensifies. Further, firms gain the diﬀerentiation benefit fromboth domestic and foreign markets. When τ falls, since it raises exports and relatively increases
the diﬀerentiation benefit gained from the foreign market, the R&D motive intensifies. Hence,if τ and k are small enough, I becomes the dominant strategy and II appears. If both costs arelarge enough, N becomes the dominant strategy and N N appears (see Fig. 1).
When τ is high and k is at an intermediate level, N N and II can appear. If firm i deviatesfrom II, wi falls. However, wj also falls due to a strategic complementarity. The deviation
makes competition tougher and the rival’s cost does not increase; hence, firm i does not deviate.
Additionally, the deviation from N N raises input price, but it brings diﬀerentiation benefit to
the deviator. However, in this case, τ is high and the size of foreign market is small, and hence
the diﬀerentiation benefit gained from the foreign market is small. Since the R&D benefit isrelatively small, there is no incentive to deviate.
Let us examine the eﬀects of τ on consumers. The formula of consumer’s surplus, CSi =1 2
22 (qii + qji + 2bqii qji ), (1), and (3) yield

CSiN N =2(2 − τ )24 − 4τ + 5τ 2; CSiII =.
8164

From (5), we establish the following.

5(5)

9329
Proposition 2. 1. Suppose k ∈ (0, 10672200]. If N N appears in N N and II, ϕ−1l (k) maximizes

consumer’s surplus.
9329
2. Suppose k ∈ ( 10672200, k). (i) If N N appears in N N and II, ϕ−1l (k) maximizes consumer’ssurplus; (ii) if II appears in N N and II, ϕ−1u (k) maximizes consumer’s surplus.
3. Suppose k ∈ [k, k). If II appear in N N and II, ϕ−1u (k) maximizes consumer’s surplus.
568
201
Here k ≡ 99225≃ 0.00572436 and k ≡ 9800
≃ 0.0205102.

Proof. See Technical Appendix.

0.09

CSiNN

0.08

0.07

0.06

CSiII
NNII

0.05

II0NN

0.057006

0.3202074τ
11

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

The input price surges if each firm invests. Since this price rise sharply increases a firm’s
production cost, the total output plummets and min CSiN N > max CSiII holds. For example,
when k < k, the equilibrium transition, N N &II→II, arises as τ decreases (see Fig. 1). In
any equilibrium, a decrease in τ raises consumer’s surplus; hence, the threshold at which theequilibrium switches N N &II to II maximizes consumer’s surplus if N N appears in N N &II(see Fig. 2).

References
Bastos, P., Straume, O., 2012. Globalization, product diﬀerentiation, and wage inequality.
Canadian Journal of Economics 45, 857–878.
6

Brander, J., Krugman, P., 1983. A ‘reciprocal dumping’ model of international trade. Journalof International Economics 15, 313–321.
Braun, S., 2008. Economic integration, process and product innovation, and relative skill
demand. Review of International Economics 16, 864–873.
Helpman, E., Krugman, P., 1985. Market Structure and Foreign Trade, MIT Press, Cambridge.
Kabiraj, T., Marjit, S., 2003. Protecting consumers through protection: the role of tariﬀinduced technology transfer. European Economic Review 47, 113–124.
Lambertini, L., Rossini, G., 1998. Product homogeneity as a prisoner’s dilemma in a duopoly
with R&D. Economics Letters 58, 297–301.
Lin, P., Saggi, K., 2002. Product diﬀerentiation, process R&D, and the nature of market
competition. European Economic Review 46, 201-211.
Rosenkranz, S., 2003. Simultaneous choice of process and product innovation when consumers
have a preference for product variety. Journal of Economic Behavior and Organization 50,183-201.
Takauchi, K., Mizuno, T., 2019. Solving a hold-up problem may harm all firms: downstream
R&D and transport-price contracts. International Review of Economics and Finance 59,29–49.

7Technical Appendix
A. Proof of Proposition 2
201
First, from Proposition 1 and its proof, II&N N appears if k < k ≡ 9800
= ϕu (0), whereas only

N N appears if k ≥ k.
In the interval (0, k), the equilibrium transition is divided into the following three types: for9329
9329], “N N &II → II” as τ decreases; for k ∈ ( 10672200, k), “N N → N N &II → II”k ∈ (0, 10672200
49329
as τ decreases; and for k ∈ [k, k), “N N → N N &II” as τ decreases. Here, ϕu ( 11) = 10672200≃
201
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’s surplus

in both the aforementioned regimes is monotonically decreasing for τ . Simple algebra yields
81, min CSiN N = CSiN N τ =4/11 = 121, and min CSiN N − max CSiII =max CSiII = CSiII τ =0 = 167
1936 > 0. These imply Proposition 2. □B. Derivation of firm i’s equilibrium profit
3rd stage: The first-order conditions for the profit maximization of firm i are ∂Πi /∂qii = 0 ⇔1 − bqji − 2qii − wi = 0 and ∂Πi /∂qij = 0 ⇔ 1 − 2qij − bqjj − wi − τ = 0 for i ̸= j. These yield
the third-stage outputs: qii (wi , wj ) =2−b+b(τ +wj )−2wi
2−b+bwj −2(τ +wi )and qij (wi , wj ) =.
4−b2
4−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 has a strategic complementarity. This is explained as follows. Supposethat supplier j raises its price. A rise in wj increases firm j’s production cost and reduces
outputs. A reduction of the rival’s outputs tends to increase firm i’s outputs; this increases theinput demand of 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 becomes as wi (b) = (2 − b)(2 − τ )/(2(4 − b)).
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 22(4 − b)2 (4 − b2 )2.

Substituting b = 1 − (dH + dF ) in Πi (b) and using the possible four combinations of dH and dF ,(dH , dF ) = {(0, 0), (0, 1/2), (1/2, 0), (1/2, 1/2)}, we obtain the equilibrium profit (4).