Strategic exporting in an international oligopoly
Kazuhiro Takauchi
Tomomichi Mizuno
December 2022
Discussion Paper No. 2216

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANStrategic exporting in an international oligopoly
Kazuhiro Takauchi∗†Tomomichi Mizuno‡

December 18, 2022
Abstract
In this paper, we build a model in which firms choose whether to export. We show that if thefixed export cost is small and the transport cost is high, the coexistence of exporters and nonexporters can appear. We also show that trade liberalization may reduce consumer and total
surpluses. Because the competition authority often cherishes consumer welfare, our finding oﬀers
an important insight into the relation between export activity and competition policy.

Key words: Exporting; Fixed export cost; Transport cost; Trade liberalization; Oligopoly
JEL classification: F12; F13∗

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.

1Introduction

In many manufacturing industries, exporters and non-exporters coexist. For example, Bernard
et al. (2007) reported that, in US data from 2002, exporters in Computer and Electronic Productwere 38% and those in Machinery Manufacturing were 33%.
It is well known that exporting is a mode of entry into a foreign market and it must havea sunk entry cost. In practice, the empirical study of Roberts and Tybout (1997) rejected theidea that the sunk entry cost is zero. This implies that firms have options regarding whether toexport, and they must pay a fixed export cost if they export their products to a foreign market.
The purpose of this paper is to build a model in which firms choose whether to export inan oligopolistic intra-industry trade framework. We show that the coexistence of exporters andnon-exporters can appear if the transport cost is high, whereas multiple equilibria in which firms
choose the same action as their rival can appear if the transport cost is low. Hence, our result ispartially consistent with empirical evidence. We also show that trade liberalization may reduce
consumer and total surpluses. The competition authority cherishes consumer welfare; hence,
our finding oﬀers an important insight related to competition policy.
In our model, firms can choose whether to export. Because the product market divides
into two when a firm exports, this paper is related to studies in which the firm’s choice of thenumber of products was considered. For example, Grossmann (2007) oﬀered a model in whichfirms decide the number of products. However, his model is limited to the domestic market, andhence, does not consider international competition.

2ModelThere are two symmetric countries: Home (labeled H) and Foreign (labeled F ). Each country
has a firm, which we call firm H and firm F , respectively. Each firm has two options, that is,whether to export or not, and pays a fixed cost k > 0 if it exports. When firm i exports, its
1output includes both domestic supply (labeled D) and exports (labeled E). We assume thatfirms have a quadratic cost function.1 The inverse demand of country i when all firms export isp = a − qii − qji , i ̸= j, i, j = H, F , where p is the product price, qii is the domestic supply offirm i, and qji is the export of firm j. The inverse demand of country i when firm j does notexport is p = a − qii .
According to the decisions made by firms, profits are as follows: When all firms choose E,their profits without k are Πi ≡ (a − qii − qji )qii + (a − qjj − qij − t)qij − 21 (qii + qij )2 fori ̸= j, i, j = H, F , where t > 0 is the per unit transport cost. When only firm i chooses D, its
profit is Πi ≡ (a − qii − qji )qii − 21 (qii )2 . When only firm i chooses E, its profit without k isΠi ≡ (a − qii )qii + (a − qjj − qij − t)qij − 12 (qii + qij )2 . When all firms choose D, their profits areΠi ≡ (a − qii )qii − 12 (qii )2 .
We consider the following two-stage game: In the first stage, each firm chooses either E orD. If a firm chooses E, it pays k. In the second stage, each firm enters a homogeneous quantity
competition. The game is solved by backward induction.

3Results

By solving the Cournot competition in the second stage of the game, we obtain the followingoutputs:
a + 2ta − 3tEE∗, qij=,552a + ta − 3tDE∗
ED∗ED∗DE∗ED∗
qHH== qFDE∗= qHF
F = qHH = qF F , qF H =77aqiiDD∗ = .
3qiiEE∗ =(1)Note that variable “EE” denotes “all firms choose E” and “DE” denotes “firm H chooses Dand firm F chooses E,” where “∗” is an equilibrium value.

1

A quadratic cost function is often used in the oligopoly model. See, for example, von Weizsäcker (1980).

2

To ensure a positive quantity, we assume t/a < 1/3.
Equation (1) yields the equilibrium profit of firms:)
1 ( 28a − 8at + 27t2 ,50
3= ΠED∗= (2a + t)2 ,ΠDE∗
HF98)
1 (ΠDE∗= ΠED∗=19a2 − 16at + 24t2 ,FH
982aΠDD∗= .
i6ΠEE∗=i(2)From equation (2), we obtain Lemma 1.
Lemma 1. (i) Suppose that rival (firm j) chooses E. If φD ≡ 2(23a−104t)(a−3t)>(≤) k/a2 , firm
1225
i chooses E (D).
4(a − 3t)2 >(≤) k/a2 , firm i chooses D (E).
(ii) Suppose that rival chooses D. If φE ≡ 147

Proof. φD ≡ ΠEE∗
− ΠDE∗= ΠEE∗− ΠED∗= 2(23a−104t)(a−3t)
and φE ≡ ΠED∗
− ΠDD∗=HHFFHH1225
4= 147− ΠDD∗(a − 3t)2 . □ΠDE∗FF

From Lemma 1, we establish Proposition 1.

Proposition 1. I. Suppose that t/a ≤ 19/162. (i) If k/a2 < φE , then EE appears. (ii) If
φE ≤ k/a2 ≤ φD , then DD&EE can appear. (iii) If k/a2 > φD , then DD appears.
II. Suppose that 19/162 < t/a < 23/104. (i) If k/a2 < φD , then EE appears. (ii) If φD ≤k/a2 ≤ φE , then DE&ED can appear. (iii) If k/a2 > φD , then DD appears.
III. Suppose that 23/104 ≤ t/a < 1/3. (i) If k/a2 ≤ φE , then DE&ED can appear. (ii) Ifk/a2 > φE , then DD appears.

The equilibrium pattern depends on the sizes of both transport and fixed costs. (See Figure1.) We first consider (a) DE&ED and then (b) DD&EE. (a) When the transport cost is highand the fixed cost is small, DE&ED can appear. If the rival chooses D, the domestic market is amonopoly. Then, because firms can gain market share in the foreign market if they export, firms
3k/a2

DD & EE
0.035

DD

0.030
0.025
0.020

φE0.0150.010

DE & ED
0.005

0.00

EE
0.05

0.10

0.15

0.20

0.25

0.30taFigure 1: Four areaschoose E. Conversely, suppose the rival chooses E. Now, the foreign market is relatively small;hence, the benefit of exporting is small. Furthermore, because the transport cost is relatively
high and the import barrier is also high, domestic market competition lessens. Hence, firms
choose D.
(b) When the transport cost is small and the fixed cost is high, DD&EE can appear.
First, suppose that there are rival exports. Now, although the fixed cost is large, the benefitof exporting is large. Furthermore, because the transport cost is low, domestic competition istougher. Then, a degree of profit reduction occurs because the foreign firm’s entry is large inthe domestic market; hence, firms choose E. Second, suppose the rival chooses D. In this case,the fixed cost is large. Because the export barrier is large and firms enjoy a domestic monopoly,
they choose D.

44Consumer surplus

Competition authorities often cherish consumer welfare compared with total surplus (e.g., Albaek, 2013); hence, we focus on consumer surplus.
From equation (1), consumer surplus in each regime isCSiEE∗ =

1a2
11DE∗
ED∗
(2a − t)2 , CSiDD∗ = , CSH= (3a − 2t)2 , CSH= (2a + t)2 ,50
189898(3)DE∗ = CS ED∗ and CS ED∗ = CS DE∗ .
where CSHFHFEquation (3) yields Proposition 2.
Proposition 2. Suppose that k/a2 < 25/2187. Then, trade liberalization with a small reductionof t may reduce consumer surplus.

DE∗ > CS EE∗ >Proof. First, DE&ED can appear if k/a2 < 25/2187. Second, ∀t/a < 1/3, CSHiED∗ . Hence, ∀k/a2 < 25/2187, consumer surplus can drop if t decreases. □CSiDD∗ > CSH

0.09
0.08

CSEE*/a2i2
CSDE*
H /a0.07
0.06

CSDD*
/a2i

0.05
0.04
0.00

2CSED*
H /at/a0.05

0.10

0.15

0.20

0.25

0.30

Figure 2: Eﬀect of trade liberalization on consumer surplus (k = 0.005)

Proposition 2 can be explained by ranking consumer surplus. (Figure 2 draws the relationshipbetween consumer surplus and transport cost.) In the regime “only the foreign firm exports,”5

because the home firm does not export, its marginal cost is small. Hence, its domestic supplyincreases. Additionally, imports exist; hence, aggregate outputs are largest among all otherregimes. Thus, consumer surplus is largest too. In the regime “everyone exports,” each firm’s
marginal cost is large. Furthermore, because strategic substitutes work in markets, the domestic
firm’s domestic supply decreases because of the rival’s exports. Hence, the aggregate output inthis regime is smaller than that in the regime “only the foreign firm exports.” In the regime “noone exports,” because there are no imports, the aggregate output in this regime is smaller thanthat in the regime “everyone exports.” Finally, in the regime “only the domestic firm exports,”
because the domestic firm supplies two markets, its marginal cost increases. Hence, its domestic
supply decreases. In this case, there are no imports; hence, the aggregate output equals thedomestic supply. As a result, consumer surplus is smallest among all other regimes.

5Total surplus

We show that trade liberalization may reduce total surplus. Total surplus in each regime is asfollows:2(3a2 − 3at + 7t2 )
23a2 − 12at + 25t2
ED∗
− k, T SH= T SFDE∗ =− k,2598
2a2
3a2 + t2DE∗
T SH
= T SFED∗ =, T SiDD∗ =.
149T SiEE∗ =By comparing the total surpluses, we obtain the following result.
Proposition 3. (i) If trade liberalization changes the trade regime, it always reduces total surplus. (ii-a) Without changing the trade regime, trade liberalization always reduces the total
surplus of the non-exporting country. (ii-b) Without changing the trade regime, the total surplusof the exporting country decreases if 3/14 < t/a < 23/104 and k/a2 < φD in the EE regime,and if 6/25 < t/a < 1/3 and k/a2 < φE in the ED regime.
Proof. See Appendix.
6To confirm that trade liberalization reduces total surplus, we drew figures with total surplus
as a function of t. In Figure 3, trade liberalization with a changing trade regime reduces totalsurplus. Additionally, the total surplus of the non-exporting country in the DE or ED regimes
decreases with t. Thus, we confirm the results (i) and (ii-a) in Figure 3.

0.230
TSEE*
/a2i0.225

TSDD*/a2i0.220
2TSDE*
H /a0.215

2
TSED*
H /a0.05

0.10

0.15

t/a0.20

Figure 3: Eﬀect of trade liberalization on total surplus (k/a2 = 0.008)

We drew the total surpluses of the exporting country in the EE and ED&DE regimes at
k/a2 = 0. Then, we obtained Figure 4. Note that from Proposition 1, either the EE or ED&DEregimes occur at k/a2 = 0. In Figure 4, the shadow areas represent the case in which tradeliberalization has a negative impact on total surplus.

0.240
0.235
0.230
0.225

TSEE*/a2i

0.220
0.215
0.00 0.05 0.10 0.15 0.20 0.25 0.30
t/aFigure 4: Welfare decreasing trade liberalization (k/a2 = 0)

7

The intuition behind Proposition 3 is as follows: As trade liberalization (decrease in t)
progresses, the exporting firm’s supply shifts from the domestic market to foreign markets. As aresult, the domestic firm’s profit increases and the domestic consumer surplus decreases. Whichof these eﬀects is larger determines whether total surplus increases. If t is large, the supply to theforeign market is small and the supply to the domestic market is large. In this case, a decrease int significantly changes the supply to the domestic market so that the eﬀect of trade liberalization
on lowering consumer surplus dominates. Thus, in each regime, trade liberalization reduces theaggregate surplus in the region in which t is large: (ii-b) in Proposition 3.
When the foreign firm trades and the domestic firm does not, trade liberalization increases
the domestic consumer surplus but reduces the domestic firm’s profit. In our model, this profitdecreasing eﬀect dominates; hence, the total surplus of the non-exporting country decreases withtrade liberalization: (ii-a) in Proposition 3.
Finally, the same intuition applies when a decrease in t changes the trade regime. When aslight decrease in t changes the trade regime, the profit of the firm that begins to export changes
little. However, the total surplus of the country in which firms begin to export falls because
the supply to the country is significantly reduced. Furthermore, because trade liberalization
intensifies competition among firms, the profits of firms that do not change their export decisions
also decline. Although consumer surplus in the country increases, the total surplus of the country
also falls because this profit-lowering eﬀect dominates: (i) in Proposition 3.

Acknowledgements
The first author’s work was supported by the Japan Society for the Promotion of Science (JSPS),
KAKENHI [grant number 20K01646]. All errors are our own.

Appendix
Proof of Proposition 38We distinguish two scenarios in which trade liberalization reduces total surplus: (i) tradeliberalization with a changing trade regime and (ii) trade liberalization without a changing trade
regime.
We consider case (i). From Proposition 1, the trade regime change occurs at k/a2 = φDor k/a2 = φE . From Proposition 1, for t/a ≤ 19/162, we compare T SiDD∗ with T SiEE∗ atk/a2 = φD or k/a2 = φE :()2(a − 3t)(109a + 93t)
T SiDD∗ − T SiEE∗ k=φ a2 => 0,
D11025( DD∗)2(a − 3t)(52a + 579t)
T Si− T SiEE∗ k=φ a2 => 0.
E
11025

Hence, for t/a ≤ 19/162, trade liberalization reduces the total surplus of both countries.
ED∗ at k/a2 = φ ,Next, we consider case (i) with t/a > 19/162. By evaluating T SiDD∗ − T SH
E

we obtain the following equation:()(a − 3t)(13a + 3t)
ED∗
T SiDD∗ − T SH=> 0.
k=φE a2882

DE∗ and T S ED∗ − T S EE∗ are
Additionally, T SiDD∗ − T SHi
HDE∗
T SiDD∗ − T SH=

(a − 3t)(a + 3t)> 0,126ED∗
T SH− T SiEE∗ =(249t − 13a)(a − 3t)
> 0.
2450

DE∗ − T S EE∗ , we obtain
Substituting k/a2 = φD into T SHi()(29a − 17t)(a − 3t)
DE∗
T SH− T SiEE∗ k=φ a2 => 0.
D2450

Therefore, trade liberalization with a changing trade regime always reduces total surplus.
We consider case (ii): trade liberalization without a changing trade regime. By diﬀerentiating
the total surplus in each trade regime with respect to t, we obtain the following:ED∗DE∗∂T SiEE∗
2(3a − 14t) ∂T SH
6a − 25t ∂T SHt∂T SiDD∗=−,=−,= > 0,= 0.
2
∂t
25a∂t
49
∂t
7
∂t

Hence, if the trade regime is ED or DE, trade liberalization always reduces the total surplusof the non-exporting country. Additionally, without the exporting country, trade liberalization

9has no eﬀect on total surplus.
Because ∂T SiEE∗ /∂t > 0 if t/a > 3/14, trade liberalization reduces total surplus if 3/14 <ED∗ /∂t > 0 if t/a > 6/25, trade liberalizationt/a < 23/104 and k/a2 < φD . Next, because ∂T SH

reduces total surplus if 6/25 < t/a < 1/3 and k/a2 < φE .
Summarizing the above results, we obtain Proposition 3.□References[1] Albaek, S., 2013. Consumer welfare in EU competition policy. Aims and Values in Competition Law, 67–88.
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