Consumer-hurting competitionin an international upstream market
Kazuhiro Takauchi
Tomomichi Mizuno
August 2023
Discussion Paper No. 2314

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANConsumer-hurting competition in an international upstream
marketKazuhiro Takauchi∗†Tomomichi Mizuno‡

August 24, 2023
Abstract
In standard oligopoly theory, an increase in the number of ﬁrms makes market competition
keener, decreases the product price, and increases consumer surplus. In practice, competition
authorities and practitioners cherish competition, and have the clear-cut belief that “competition enhances consumer welfare.” However, we oﬀer a diﬀerent perspective on the eﬀect ofcompetition on consumer welfare. In vertically related markets with an import tariﬀ, we showthat an increase in the number of upstream ﬁrms can reduce the consumer surplus and total
surplus of the ﬁnal-good importing country. Upstream competition may harm consumers.

Key words: Upstream competition; Import tariﬀ; Consumer surplus; Vertically related markets
JEL classiﬁcation: F12; F13; L13∗

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.
†
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

Competition authorities and practitioners undoubtedly cherish competition, and have the clearcut belief that “competition enhances consumer welfare.” For example, Joaquı́n Almunia1 saidon the importance of competition: “Our objective is to ensure that consumers enjoy the beneﬁts
of competition, a wider choice of goods, of better quality and at lower prices. But competition
does not only deliver beneﬁts for consumers. It also delivers beneﬁts for business and theeconomy as a whole” (Competition and consumers: The future of EU competition policy, 2010,p. 4). This assertion is not baseless. In standard oligopoly theory, market power decreases as
the number of ﬁrms increases, and the higher the number of ﬁrms, the lower the prices and thehigher consumer surplus (Motta, 2004, p. 51). Hence, we could consider that the promotion ofcompetition caused by an increase in the number of ﬁrms enhances consumer welfare.
In this paper, we oﬀer a diﬀerent perspective against such a belief. When we consider an
import tariﬀ in a vertically related market, an increase in the number of upstream input suppliers
may reduce consumer surplus and total surplus; that is, the promotion of competition does notalways enhance consumer welfare.
We consider an international vertical production structure that comprises n upstream input
suppliers that engage in homogeneous price competition and two downstream ﬁnal-good exporting ﬁrms that supply diﬀerentiated goods. These two ﬁnal-good exporting ﬁrms are locatedin a foreign exporting country without a ﬁnal-good market and pay a tariﬀ when they supply
their products to the ﬁnal-good importing country. Upstream, m input suppliers belong to theﬁnal-good importing country, and n − m input suppliers belong to a country other than theﬁnal-good importing country and ﬁnal-good exporting country. In this environment, we showthat if the number of input suppliers n increases in the case m > 1, the consumer and total
surpluses of the ﬁnal-good importing country decrease.
1

Vice-president and commissioner for competition, European Commission 2010–2014.

1

This result depends on an increase in the unit cost of the ﬁnal-good exporting ﬁrms. The
unit cost of the ﬁnal-good exporting ﬁrm equals the sum of the input price and tariﬀ. An
increase in the number of input suppliers lowers the input price, whereas it tends to raise thetariﬀ level. When m > 1, because the tariﬀ rising eﬀect is dominant, the unit cost of the ﬁnalgood exporting ﬁrm increases as the number of input suppliers increases. This makes ﬁnal-good
exports lower, and hence, consumer surplus smaller. Furthermore, because the reduction eﬀectof the consumer surplus is dominant, the total surplus in the ﬁnal-good importing country also
decreases.
We extend the benchmark model to two cases. One is the case in which the upstream
market is a homogeneous quantity competition and the other is the case in which a downstream
oligopolist exists in the ﬁnal-good importing country. In both two cases, we obtain a similar
result to that obtained in the benchmark case. Hence, we can claim that our model has certain
robustness.
It tends to be considered that the promotion of competition enhances consumer welfare;however, we prove that this is not always true under vertically related markets. This point is
a contribution of our study. Additionally, it is important that the promotion of competition
can hurt consumers in a realistic case such that ﬁnal-good producers purchase inputs from aninternational market. Our result depends on the optimal tariﬀ policy of the ﬁnal-good importing
country. Hence, it may be necessary for competition authorities to confer with their governments
to ease the negative eﬀect of the tariﬀ policy on consumer welfare.
Our paper is related to two strands in the literature. One is for studies on the relationshipbetween competition and consumer welfare (Deltas et al., 2012; Dinda and Mukherjee, 2014;
Mukherjee and Sinha, 2019; and Wang and Mukherjee, 2012) and the other is for studies on theeﬀects of the optimal tariﬀ under a vertical production chain (Ara and Ghosh, 2016; Ishikawa
and Lee, 1997; and Lahiri and Ono, 1999).2 Using the Hotelling model, Deltas et al. (2012)2

Moreover, Takauchi (2011) considered the tariﬀ policy of a ﬁnal-good importing country within a free trade2

showed that collusion (i.e., restriction of competition) can beneﬁt consumer welfare. Dinda andMukherjee (2014) demonstrated that an increase in the number of ﬁrms can reduce consumer
surplus under the optimal tax/subsidy policy, quoting empirical evidence such that a new entry
increases the price and reduces consumer welfare. Mukherjee and Sinha (2019) showed that, ina third-market model, a cartel between two exporting ﬁrms may enhance consumer welfare in
the third market. Wang and Mukherjee (2012) showed that, in a nationalized monopoly market,
the entry of private ﬁrms hurts consumers. Although these researchers respectively considered
the relationship between competition and consumer welfare using diﬀerent models, they did notconsider an upstream market. Diﬀerent from these researchers, we consider an upstream market.
Ara and Ghosh (2016) considered a tariﬀ policy in the case in which a foreign country specializes in intermediate-good production and the home country specializes in ﬁnal-good production.
Ishikawa and Lee (1997) examined the eﬀects of tariﬀs in both upstream and downstream markets. Lahiri and Ono (1999) considered an optimal tariﬀ policy when producers and sellers are
diﬀerent. Although these researchers considered vertical production structures, they did notconsider price competition in the upstream market; hence, their models diﬀer substantially fromours.
This paper is organized as follows: In Section 2, we oﬀer a benchmark model and in Section
3, we present the analysis results. In Section 4, we discuss and extend the benchmark model.
In Section 5, we conclude the paper.

2

Model

In the model, two countries exist: ﬁnal-good importing and exporting countries. Hereafter, wecall the ﬁnal-good importing country the ﬁnal-good importer. Here, for better understanding,the ﬁnal-good importer has no ﬁnal-good producers and mainly imports foreign ﬁnal goods. In

area.

3

Section 4, we relax this assumption and argue the case in which the ﬁnal-good importer has
a ﬁnal-good oligopolist. The ﬁnal-good importer imposes a common trade policy, t, on theﬁnal-good exporting ﬁrms (e.g., if t > 0, it denotes a speciﬁc tariﬀ; and if t < 0, it denotes animport subsidy). The ﬁnal-good exporting country has no ﬁnal-good markets; however, it has
two ﬁnal-good exporting ﬁrms: A and B. These ﬁrms export horizontally diﬀerentiated goods
to the ﬁnal-good importer and they face the trade policy.
In the upstream market, n(> 2) input suppliers exist and they engage in a homogeneous pricecompetition: m input suppliers belong to the ﬁnal-good importer, and n − m input suppliers
are located in a country other than the ﬁnal-good importer and ﬁnal-good exporting country.
To produce one unit of the ﬁnal good, ﬁrms A and B use one unit of input. The inverse market
demand function of the ﬁnal-good importer is given by pi = 1 − qi − bqj , i = j, i, j = A, B,
where pi and qi are the price and quantity supplied by ﬁrm i (i = A, B), respectively, andb ∈ [0, 1) represents the degree of product substitutability. The proﬁt of ﬁrm i (i = A, B) isΠi ≡ (pi − r − t)qi , where r is the input price. For simplicity, we omit transportation costs.
Let the input price oﬀered by supplier k (k ∈ {1, . . . , n}) be rk , supplier k’s individual
demand be xk , and total demand be qA + qB . Because each ﬁnal-good exporting ﬁrm purchases
the lowest input, the individual demand of input supplier k is xk = [qA (rl ) + qB (rl )]/h if the
supplier oﬀers the lowest price rk = rl . Note that, h is the number of input suppliers thatoﬀer the lowest price. When input supplier k oﬀers a higher price than rl , its demand is zero:xk = 0. To obtain explicit solutions, we assume that the production cost of input supplier k is(λ/2)x2k , where λ > 0 is production eﬃciency.3 The proﬁt of input supplier k (∈ {1, . . . , n}) is
πk ≡ rk xk − (λ/2)x2k .4

3This type quadratic cost function was frequently used in previous studies. See, for example, the studies by
Dastidar (1995), Delbono and Lambertini (2016b), Gori et al. (2014), Mizuno and Takauchi (2020, 2022), andTakauchi and Mizuno (2022).
4
In the real world, several manufacturing ﬁrms’ technology can be decreasing returns to scale. For example,
using the aggregate data of 34 manufacturing industries in the U.S., Basu and Fernald (1997) found that a typicalindustry appears to have decreasing returns to scale.

4We consider the following three stage game. In the ﬁrst stage, the ﬁnal-good importerchooses the level of trade policy t. In the second stage, each input supplier k decides its price.
In the third stage, the ﬁnal-good exporting ﬁrm decides its exports. Because multiple Nashequilibria (range of price) appear in the second stage of the game, we use the subgame perfectNash equilibrium with the payoﬀ-dominance criterion as the equilibrium concept.5 The game issolved by backward induction.

3

Results

In the third stage of the game, each ﬁnal-good exporting ﬁrm decides its export to maximize
proﬁt. The ﬁrst-order conditions (FOCs) for proﬁt maximization are 1 − 2qA − bqB − r − t = 0
and 1 − bqA − 2qB − r − t = 0. These FOCs yield the following third-stage export:qi (r, t) =1−r−t
, i = A, B.
2+b

In the second stage, input suppliers decide their prices. According to Dastidar (1995),6 ifoligopolists have a convex cost, the Nash equilibria in a homogeneous price competition among
them have an interval [r, r]. The lower input price r is derived from the following condition:πk (r, t; n) ≡ r

qA (r, t) + qB (r, t)
n



λ
−
2



qA (r, t) + qB (r, t)
n

2

≥ 0.

The upper input price r is derived from the following condition:πk (r, t; n) ≥ πk (r, t; 1) ≡ r (qA (r, t) + qB (r, t)) −λ(qA (r, t) + qB (r, t))2 ,2where πk (r, t; 1) is the monopoly proﬁt of input supplier k.
Furthermore, the collusive input price rcol (which maximizes industry proﬁt) is derived from

5This concept is often used. For example, see the studies by Cabon-Dhersin and Drouhin (2014, 2020), Mizuno
and Takauchi (2020, 2022), and Takauchi and Mizuno (2022).
6
Dastidar (1995)-type price competition is often used in many studies. See, for example, the studies by CabonDhersin and Drouhin (2014, 2020), Delbono and Lambertini (2016a, b), Gori et al. (2014), Mizuno and Takauchi(2020, 2022), and Takauchi and Mizuno (2022).

5

argmaxr πk (r, t; n). These prices becomesr=λ(1 − t)λ(n + 1)(1 − t)
(1 − t)((b + 2)n + 2λ), r=, rcol =.
(b + 2)n + λn(b + λ + 2) + λ2((b + 2)n + λ)(1)

The above (1) yields Lemma 1.
Lemma 1. (i) r < r. (ii) rcol > r iﬀ λ <(2 + b)n.
n−12(b+2)λn (1−t)
Proof. (i) Simple algebra yields r − r = (bn+λ+2n)(bn+λ+λn+2n)> 0. (ii) From (1), we obtainrcol − r =(b + 2)n(1 − t)(bn + λ − λn + 2n).
2(bn + λ + 2n)(bn + λ + λn + 2n)

By solving rcol − r > 0 with respect to λ, we obtain Part (ii). To ensure rcol > r, we require the following.
Assumption 1. λ <(2 + b)n.
n−1

In the ﬁrst stage, the ﬁnal-good importer decides t to maximize its total surplus:
W ≡ CS +mπk + t(qA + qB ),

(2)

k=1
2 +2bq q +q 2 )/2 is consumer surplus. From (2) and the second-stage outcomes,
where CS = (qA
A B
B

the optimal import policy t∗ becomest∗ =λ(1 − 2m + n) + n.
n(b + 2λ + 3) − 2λ(m − 1)(3)

Note that “∗” denotes the equilibrium outcome.
To ensure positive equilibrium values, we make the following assumption.7
Assumption 2. m < m0 ≡2λ + n(b + 2λ + 3).
2λThe ﬁnal-good export is
qi∗ =
7

n
> 0, i = A, B.
n(b + 2λ + 3) − 2λ(m − 1)

The second-order (suﬃcient) condition for welfare maximization is satisﬁed provided Assumption 2 holds.

6

The proﬁt of the ﬁnal-good exporting ﬁrm is Π∗i = (qi∗ )2 .
The input price isr∗ =λ(n + 1).
n(b + 2λ + 3) − 2λ(m − 1)

(4)

The proﬁt of input supplier k is
πk∗ =

2λn
for all k ∈ {1, . . . , n}.
[n(b + 2λ + 3) − 2λ(m − 1)]2

From (3), we establish Proposition 1.

Proposition 1. (i) Suppose that m < (λ + λn + n)/2λ. The optimal trade policy of the ﬁnalgood importer is a tariﬀ, that is, t∗ > 0. (ii) Suppose that m = (λ + λn + n)/2λ. The optimal
trade policy of the ﬁnal-good importer is free trade, that is, t∗ = 0. (iii) Suppose that m >(λ + λn + n)/2λ. The optimal trade policy of the ﬁnal-good importer is an import subsidy, thatis, t∗ < 0.
Proof. From (2), the critical value t∗ = 0 is m = (λ + λn + n)/2λ. This value is smaller thanm0 , that is, m0 − (λ + λn + n)/2λ = (2n + bn + λ + nλ)/2λ > 0. Hence, Proposition 1 holds. The logic behind Proposition 1 is as follows: When the number of domestic input suppliers
m is large, it is important for the ﬁnal-good importer to expand input demand. Because thesize of input demand depends on the volume of the ﬁnal-good exports, it is optimal to enhance
the proﬁt of the domestic input suppliers by expanding input demand because of the increaseof ﬁnal-good exports. Hence, to improve ﬁnal-good exports, the ﬁnal-good importer oﬀers animport subsidy.
When the number of domestic input suppliers is small, input demand is not important forthe ﬁnal-good importer. Because the proﬁt of the input supplier is relatively small comparedwith the total surplus, it is not desirable from the welfare viewpoint to protect input suppliers.
Hence, it is optimal to impose a tariﬀ on the ﬁnal-good exporting ﬁrms to gain tariﬀ revenue.

7

Lemma 2. (i) The optimal trade policy of the ﬁnal-good importer, t∗ , decreases as m increases.
(ii-a) If m = 0, t∗ decreases as n increases. (ii-b) If m ≥ 1, t∗ increases as n increases.

Proof. By diﬀerentiating (3) with respect to m and n, we obtain
2λ(n(b + λ + 2) + λ)
∂t∗
=−
< 0,
∂m
[n(b + 2λ + 3) − 2λ(m − 1)]2
λ(b(2m − 1) + 2(λ + 2)m − 1)
∂t∗
=
.
∂n
[n(b + 2λ + 3) − 2λ(m − 1)]2
These imply Lemma 2. Lemma 3. (i) The input price, r∗ , increases as m increases. (ii) r∗ decreases as n increases.

Proof. By diﬀerentiating (4) with respect to m and n, we obtain
2λ2 (n + 1)
∂r∗
=
> 0.
∂m
[n(b + 2λ + 3) − 2λ(m − 1)]2
λ(b + 2λm + 3)
∂r∗
=−
< 0.
∂n
[n(b + 2λ + 3) − 2λ(m − 1)]2
Hence, Lemma 3 holds. 
We consider that m increases. Then, the ﬁnal-good importer attempts to protect its domestic
input suppliers through promoting ﬁnal-good imports, that is, by enhancing input demand.
Hence, the level of trade policy t∗ decreases. Because input suppliers raise their prices according
to the expanded input demand, the input price increases and the proﬁt of the input suppliersalso increases.
When n increases, because the international upstream market becomes more competitive,
input price r falls ((ii) of Lemma 3). By contrast, this reduction in the input price reduces the
proﬁt of the input suppliers. Therefore, if the input supplier exists in the ﬁnal-good importer,
its total surplus can decrease because of the increase of n. Then, to prevent a decrease in the
total surplus, the ﬁnal-good importer attempts to gain revenue from the trade policy through arise in the level of trade policy. Therefore, t∗ increases as n increases ((ii) of Lemma 2).

8

By substituting equilibrium outcomes into the deﬁnition of consumer surplus and (2), we
obtain equilibrium consumer and total surpluses:
CS ∗ =(b + 1)n2n.
, W∗ =[n(b + 2λ + 3) − 2λ(m − 1)]2n(b + 2λ + 3) − 2λ(m − 1)(5)

From (5), we establish Proposition 2.
Proposition 2. (i) If m = 0, consumer and total surpluses increase as n increases. (ii) If
m = 1, consumer and total surpluses do not change as n increases. (iii) If m > 1, consumer
and total surpluses decrease as n increases.
Proof. By diﬀerentiating (5) with respect to n, we obtain∂CS ∗4(b + 1)λ(m − 1)n=−,∂n[n(b + 2λ + 3) − 2λ(m − 1)]32λ(m − 1)
∂W ∗
=−
.
∂n
[n(b + 2λ + 3) − 2λ(m − 1)]2
These imply Proposition 2. To consider Proposition 2, we set Lemma 4.
Lemma 4. (i) If m = 0, the unit cost of the ﬁnal-good exporting ﬁrm, r∗ + t∗ , decreases as nincreases. (ii) If m = 1, r∗ + t∗ does not change as n increases. (iii) If m > 1, r∗ + t∗ increases
as n increases.
Proof. By diﬀerentiating r∗ + t∗ with respect to n, we obtain2(b + 2)λ(m − 1)∂(r∗ + t∗ )=.
∂n[n(b + 2λ + 3) − 2λ(m − 1)]2This implies Lemma 4. 
If the unit cost of the ﬁnal-good exporting ﬁrms, r∗ + t∗ , increases because of an increasein the number of input suppliers n, ﬁnal-good exports decrease (Lemma 4). Then, because
ﬁnal-good exports decrease, consumer surplus decreases. The eﬀect of a decrease in consumer
surplus is dominant; hence, the total surplus also decreases (Proposition 2).
9We consider the “m = 0” case. The level of trade policy decreases and the input price falls
because of an increase in the number of input suppliers (Lemmas 2 and 3). Hence, the unit costof ﬁnal-good exporting ﬁrms always decreases as n increases.
When m = 1, the level of trade policy increases as n increases. Because n increases, theupstream market becomes competitive and the input price lowers. Then, each input supplier’s
proﬁt also decreases. By raising the burden of the trade policy, the ﬁnal-good importer tries
to compensate for the proﬁt reduction of its domestic input supplier. In this case, because thereduction of the input price is canceled out by the increase in the level of trade policy, the unit
cost does not change, even if n increases. Hence, an increase in n does not aﬀect consumer andtotal surpluses.
When m > 1, because the eﬀect of an increase in the level of trade policy dominates theeﬀect of a fall in the input price due to an increase in n, the unit cost increases as n increases.
Therefore, an increase in n reduces both consumer and total surpluses.

4

Discussion and extension

4.1

Upstream quantity competition

Does the upstream competition mode change our main result? To answer this, we introduce
homogeneous quantity competition in the upstream market. With the exception of quantity
competition in the upstream market, the model is the same as that in the previous section.
From the third-stage exports of ﬁnal-good exporting ﬁrms, qi (r, t) for i = A, B, we obtainqA (r, t) + qB (r, t) = Y =n

k=1 yk , where yk

is the output of each input supplier k and Y

denotes the aggregate outputs. By solving qA (r, t) + qB (r, t) = Y with respect to r, we obtainr = [2 − 2t − (2 + b)Y ]/2. Because the proﬁt of input supplier k (∈ {1, . . . , n}) is πkC ≡(1−t)(b+2c+2). Using r(t) andryk − (c/2)yk2 , the second-stage input price becomes r(t) = b(n+1)+2(c+n+1)

10

second-stage outcomes, the welfare maximizing trade policy becomestC∗ =n(b + 2c + n + 2) − 2m(b + c + 2),n(b(n + 2) + 4c + 3n + 4) − 2m(b + c + 2)

(6)

where “C” denotes upstream Cournot competition.
From (6), Proposition 3 holds.
Proposition 3. Suppose that the international upstream market is a homogeneous quantity
competition. I. Let n < 3. (i) If m < n(b+2c+n+2)
2(b+c+2) , the optimal trade policy of the ﬁnal-good
importer is a tariﬀ, that is, tC∗ > 0. (ii) If m = n(b+2c+n+2)
2(b+c+2) , the optimal trade policy of the
ﬁnal-good importer is free trade, that is, tC∗ = 0. (iii) If m > n(b+2c+n+2)
2(b+c+2) , the optimal trade
policy of the ﬁnal-good importer is an import subsidy, that is, tC∗ < 0.
II. Let n > 3. The optimal trade policy of the ﬁnal-good importer is a tariﬀ, that is, tC∗ > 0.
Proof. From (6), by solving the inequality tC∗ ≥ 0 with respect to m, we obtain m ≤ n(b+2c+n+2)
2(b+c+2) .
C∗ < 0 can appear if n < 3. These imply Proposition 3.
= (2+b−n)nBecause n − n(b+2c+n+2)
2(b+c+2)2(c+b+2) , tThe logic behind Proposition 3 is relatively intuitive. The relaxation in upstream competition
gives the ﬁnal-good importer the incentive to protect domestic input suppliers through expanding
input demand weaker. Hence, for the ﬁnal-good importer, the incentive to gain tariﬀ revenue
becomes stronger. As a result, the ﬁnal-good importer can oﬀer an import subsidy only when
the number of upstream ﬁrms is small.
The equilibrium input price isrC∗ =n(b + 2c + 2).
n(b(n + 2) + 4c + 3n + 4) − 2m(b + c + 2)

11

(7)

The equilibrium welfares are
CS C∗ =(b + 1)n4,[n(b(n + 2) + 4c + 3n + 4) − 2m(b + c + 2)]2

W C∗ =n2.
n(b(n + 2) + 4c + 3n + 4) − 2m(b + c + 2)

(8)

From (8), we establish Proposition 4.
Proposition 4. Suppose that the international upstream market is a homogeneous quantity
competition. (i) If m > n(b+2c+2)2(b+c+2) , consumer and total surpluses decrease as n increases. (ii)
If m = n(b+2c+2)2(b+c+2) , consumer and total surpluses do not change as n increases. (iii) If m <n(b+2c+2)2(b+c+2) , consumer and total surpluses increase as n increases.

Proof. By diﬀerentiating (8) with respect to n, we obtain4(b + 1)n3 [n(b + 2c + 2) − 2m(b + c + 2)]∂CS C∗=,∂n[n(b(n + 2) + 4c + 3n + 4) − 2m(b + c + 2)]32n[n(b + 2c + 2) − 2m(b + c + 2)]
∂W C∗
=
.
∂n
[n(b(n + 2) + 4c + 3n + 4) − 2m(b + c + 2)]2
From the numerator of the equations, sign{∂CS C∗ /∂n} = sign{∂W C∗ /∂n} = sign{n(b + 2c +n(b+2)C∗ /∂n <(>)0 if
2) − 2m(b + c + 2)}. Moreover, n − n(b+2c+2)2(b+c+2) = 2(b+c+2) > 0. Hence, ∂CSm >(<) n(b+2c+2)2(b+c+2) . 
In the upstream quantity competition, we also ﬁnd a similar feature of the unit cost in theﬁnal-good exporting ﬁrms with price competition. From (6) and (7), we ﬁnd2(b + 2)n[2m(b + c + 2) − n(b + 2c + 2)]∂(rC∗ + tC∗ )=.
∂n[n(b(n + 2) + 4c + 3n + 4) − 2m(b + c + 2)]2This equation yields the critical value n(b+2c+2)2(b+c+2) , and hence, we ﬁnd a similar comparative statics
result with price competition in the welfares (Proposition 4).

12

4.2

Final-good importer with a downstream oligopolist

We introduce a downstream ﬁrm located in the ﬁnal-good importer in our model. For simplicity,
we consider homogeneous quantity competition (i.e., b = 1) in the ﬁnal-good market. Hence,
the inverse market demand of the ﬁnal good is p = 1 − qH − qA − qB . We assume that λ <8n/(3n − 3) ≡ λ̄ho , which guarantees that the collusive input price is larger than the highest
price in the Nash equilibria in the upstream market. Additionally, we suppose that m < (2nλ +nλ)(10n + λ + nλ)/(4nλ) ≡ mho , which is the second-order condition in the ﬁrst stage.
By applying a similar calculation to that in the previous section, we derive the optimal trade
policy:t∗ho =

n[−6λm + 6n + 5λ(1 + n)]
λ2 + 2λn(λ − 2m + 6) + (λ2 + 12λ + 20) n2.

By considering the sign of t∗ho , we obtain the following proposition.
Proposition 5. Suppose that one downstream ﬁrm exists in the importing country and all downstream ﬁrms produce a homogeneous product and compete in quantity. (i) If m < 6n+5λ(1+n)≡
6λmtho , the optimal trade policy of the ﬁnal-good importer is a tariﬀ, that is, t∗ho > 0. (ii) If
m = mtho , the optimal trade policy of the ﬁnal-good importer is free trade, that is, t∗ho = 0. (iii)
If m > mtho , the optimal trade policy of the ﬁnal-good importer is an import subsidy, that is,t∗ho < 0.
Proof. The sign of t∗ho only depends on the terms in the numerator: −6λm + 6n + 5λ(1 + n).
Note that the denominator is positive because the second-order condition in the ﬁrst stage mustbe satisﬁed. By solving −6λm + 6n + 5λ(1 + n) > 0 for m, we obtain the following inequality:m<

6n + 5λ(1 + n)
≡ mtho .
6λAdditionally, a comparison of mtho and mho always yields mho > mtho , which implies Proposition
5.

13

Taking into account the presence of a downstream ﬁrm located in the importing country, theﬁnal-good importer chooses to impose a tariﬀ when the number of input suppliers located in theﬁnal-good importer is small. Because this result is consistent with Proposition 1, the intuition
behind both is shared.
Next, we consider the eﬀect of n on the consumer and total surpluses. In equilibrium,
consumer and total surpluses are as follows:
∗
CSho
=
∗
Who
=

2n2 (λ + (λ + 6)n)2[λ2 + 2λn(λ − 2m + 6) + (λ2 + 12λ + 20)n2 ]22n(λm + 4n)
λ2 + 2λn(λ − 2m + 6) + (λ2 + 12λ + 20) n2,.

By diﬀerentiating these surpluses with respect to n, we obtain the following derivatives:
 





∗
4λn(λ + (λ + 6)n) m −4λn2 − 24n2 + λ2 n2 + 2n + 1 + λ 12n2 + 12n + 52n2
∂CSho
=
,
∂n
[λ2 + 2λn(λ − 2m + 6) + (λ2 + 12λ + 20) n2 ]3∗∂Who2λ(λ + (λ + 6)n)[−λm(n − 1) − 2n(3m − 4)]
=
< 0.
∂n
[λ2 + 2λn(λ − 2m + 6) + (λ2 + 12λ + 20) n2 ]2

We deﬁne the following threshold values:
λ
mCS
ho ≡

2+

 2

λ + 12λ + 52 n2 + 2λ(λ + 6)n,4(λ + 6)n2

mW
ho ≡

8n
.
(λ + 6)n − λThen, we obtain the result of the comparative statics as follows.
Proposition 6. Suppose that one downstream ﬁrm exists in the importing country and all
downstream ﬁrms produce a homogeneous product and compete in quantity. (i) Consumer surplus
Wdecreases with n if m > mCS
ho ; and total surplus decreases with n if m > mho . (ii) ConsumerW
surplus does not change with n if m = mCS
ho ; and total surplus does not change with n if m = mho .

(iii) Consumer surplus increases with n if m < mCS
ho ; and total surplus increases with n if
m < mWho .
∗ . From the second-order condition in the ﬁrst
Proof. First, we consider the eﬀect of n on CSho
∗ /∂n is positive. Then, the sign of ∂CS ∗ /∂n only depends onstage, the denominator of ∂CShoho14∗ /∂n < 0 for m and obtain m > mCS . By
the terms in the numerator. Hence, we solve ∂CShohocomparing mCS
ho and mho , we obtain the ﬁrst part of the proposition.
∗ . The sign of ∂W ∗ /∂n only depends on the terms
Next, we consider the eﬀect of n on Whoho∗ /∂n < 0 for m and obtain m > mW . By comparing
in the numerator. Hence, we solve ∂WhohomW
ho and mho , we obtain the second part of the proposition. We can conﬁrm that Proposition 6 is consistent with Proposition 2. Thus, the intuition
behind Proposition 6 is the same as that behind Proposition 2.

5

Conclusion

In this paper, we considered a ﬁnal-good importer with upstream input suppliers, and showedthat upstream competition can reduce consumer and total surpluses in the ﬁnal-good importer.
In international upstream markets, the input price falls as the number of input suppliers increases. By contrast, if the input supplier exists in the ﬁnal-good importer, the level of its
trade policy increases as the number of input suppliers increases. The unit cost of the ﬁnalgood exporting ﬁrms is the sum of the level of trade policy and input price. When the numberof input suppliers increases, the input price falls, but the level of trade policy rises. Hence,an increase in the number of input suppliers has conﬂicting eﬀects. If the ﬁnal-good importer
has an input supplier, as the cost-increasing eﬀect is dominant, an increase in the number of
input suppliers increases the cost of ﬁnal-good exporting ﬁrms and reduces their export volumes. Because imports of the ﬁnal good decrease and the price increases, consumer surplus
decreases. Furthermore, the ﬁnal-good exporting ﬁrms become less eﬃcient; hence, the total
surplus also decreases. When (i) upstream competition is quantity and (ii) an oligopolist exists
in the ﬁnal-good importer, this result essentially holds.
Competition authorities and practitioners usually emphasize competition. However, it is notnecessarily desirable for consumers and the entire economy to enhance competition rashly. Our
15contribution is that we found that an increase in the number of upstream ﬁrms (competition
enhancing) may cause such an undesirable result.
Our analysis is limited to the “one importing country case.” Therefore, extending our model
to a two-way two-country trade model may be fruitful. However, this topic is beyond the scopeof our analysis and remains an issue for future research.

Acknowledgements
We thank Noriaki Matsushima for his helpful comments. We thank Edanz (https://jp.edanz.com/ac)
for editing a draft of this manuscript. All errors are our own.

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