Optimal export policy with upstream pricecompetition
Tomomichi Mizuno
Kazuhiro Takauchi

February 2018
Discussion Paper No.1805

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANOptimal export policy with upstream price competition
Tomomichi Mizuno∗
Graduate School of Economics, Kobe University
Kazuhiro Takauchi†
Faculty of Business and Commerce, Kansai University
February 8, 2018
Abstract
We constructed a third-market model with a vertical trading structure in which input
suppliers engage in the homogeneous price competition à la Dastidar (1995). We showthat in the case of downstream Bertrand competition, a non-monotonic export policy
may appear, that is, the optimal export policy can change like a tax–subsidy–tax as
the degree of product-substitutability rises. We also show that when the number ofdomestic input suppliers is at an intermediate level, the conventional result in which
the optimal policy is an export subsidy (tax) if downstream is Cournot (Bertrand)
rivalry remains. We further discuss welfare comparisons between downstream Cournot
and Bertrand cases.
Key words: Upstream price competition; Export subsidy/tax; Non-monotonic policy;
Product substitutability
JEL classification: F12; F13; L13; D43∗

Graduate School of Economics, Kobe University, 2-1 Rokkodai-cho, Nada-ku, Kobe-City, Hyogo
657-8501, Japan. E-mail: mizuno@econ.kobe-u.ac.jp; Tel.: +81-78-803-7245; Fax: +81-78-803-7289.
†
Corresponding author: Faculty of Business and Commerce, Kansai University, 3-3-35 Yamate-cho,Suita, Osaka 564-8680, Japan. E-mail: kazh.takauch@gmail.com; Tel.: +81-6-6368-1817; Fax: +81-66339-7704.

1

Introduction

Vertical trade links are a prominent feature of modern international trade. As production fragmentation rises, each country specializes in producing particular inputs or ina production stage, and the vertical trading chain reaches many countries (Hummels
et al., 2001). Additionally, the progress of trade liberalization, such as the growing
economic integration and a reduction in trade costs enables firms to purchase inputs
internationally. With progress in the globalization in production, the use of importedinputs has been expanding and trade in inputs plays a central role in goods trade (Aliand Dadush, 2011; World Trade Organization (WTO), 2009, 2013). According to theWTO (2009, 2013), trade in intermediate inputs (excluding fuel) occupied 40% of total
trade in 2008, and the share of intermediate inputs in non-fuel exports was over 50%
during 2000–2011.
Considering vertical trade forced the conventional argument on trade policies tochange. Such a change of argument is outstanding in discussions on export policy
because when the input market is imperfectly competitive, the rent-shifting eﬀect ofexport subsidies/taxes is due not only to foreign downstream rivals, but also upstream
input-suppliers. Bernhofen (1997) shows that when a monopoly input-supplier exists
outside the country, an export subsidy makes input demand less elastic and enables
the monopoly supplier to set a higher input price, so the government’s incentives tosubsidize weakens.1 Although Bernhofen (1997) sheds light on the fact that a horizon1

Some studies emphasize diﬀerent factors that aﬀect the recommended export policy. Ishikawa andSpencer (1999) employ a more general setting in which upstream and downstream markets consist of
many firms, and show that the government’s decision of whether to subsidize or tax is influenced by the
number of firms. Chang and Sugeta (2004) consider Nash-bargaining between an upstream monopolist

1tal (vertical) rent-shifting eﬀect points to a subsidy (tax), it assumes a homogeneous
Cournot duopoly in the third market. In contrast, Chou (2011) extends Bernhofen’s
model to a diﬀerentiated duopoly and demonstrates that product substitutability affects the optimal export policy. In a diﬀerentiated Cournot duopoly, if the degree of
product substitutability is small and the upstream monopolist has uniform pricing,horizontal competition is gentle and the vertical rent-shifting eﬀect dominates the horizontal one. Then, the optimal export policy can become a tax, despite Cournot rivalry.
Moreover, input market integration is also important. Kawabata (2010) shows that ina third-market model with a diﬀerentiated duopoly, if each exporter has an input supplier and the input market is integrated, the horizontal rent-shifting eﬀect dominates
the vertical one, and thus, the recommended export policy can be a subsidy, despite
downstream Bertrand rivalry.2
Existing works demonstrate that in vertical trade, horizontal and vertical rentshifting eﬀects due to export policies can be influenced by market structures, productsubstitutability, and price discrimination in inputs; hence, they provide new insights
into the conventional wisdom on export policy.3 However, they assume that the upstream input market is a monopoly or input suppliers engage in quantity competition,
so the implications of price competition in the input market tend to be overlooked.
When one recognizes the fact that a Cournot industry is not observed so much in theand downstream firms in a diﬀerentiated duopoly, and show that bargaining power aﬀects export
policies. Hwang et al. (2007) demonstrate that the degree of the returns to scale of the production
function for the downstream monopolist aﬀects the government’s decision in the case of upstream
monopoly.
2
In a similar model setting, Kawabata (2012) focuses on the role of cost asymmetries among downstream firms and considers export policies for both upstream and downstream firms.
3
Takauchi (2010) also examines the policy interaction between export subsidy/tax and requirements
for the rules of origin in a trade model with a vertical production structure.

2

real world,4 it is significant to consider upstream price competition.
This paper considers the implications of upstream price competition for the recommended export policy. To achieve this aim, in a third market model with diﬀerentiatedproducts, we incorporate Dastidar (1995)-type price competition into the input market, and show that if the downstream has Bertrand rivalry, the optimal export policy
can be tax–subsidy–tax, corresponding to the degree of product substitutability. In
the downstream Bertrand case, the domestic firm’s exports are U-shaped for productsubstitutability. Hence, a rise in the degree of substitutability when it is at a low level
reduces the domestic firm’s exports and worsens its competitive position, such that it
weakens the incentives for taxation. Conversely, a high degree of substitutability raises
domestic firm’s exports, and incentives for taxation become strong. The optimal export policy has a “tax–subsidy–tax” shape, so it becomes a tax when substitutability
is both low and high. When the actual level of substitutability is low, though practitioners accidentally recognize that the degree of substitutability is high, because therealized policy is a tax, the welfare loss may be small. In contrast, if practitioners
recognize that the degree of substitutability is slightly higher than its actual level, thepolicy opposite to the optimal policy exists and the welfare loss may be considerable.
This implies that in implementing export policies, a case exists in which “a large error
is permissible, while a small error is not allowable,” and indicates that the common
knowledge that “great mistakes are impermissible” is not always true. We believe that
our result gives a new insight into the context of trade policy.
4

For example, using annual data during 1961–90 for seventy Japanese manufacturing industries,
Flath (2012) empirically shows that whereas the Cournot specification is the most likely for 5 industries,
the Bertrand specification is the most likely for 35 industries.

3

We also show that the conventional results in Brander and Spencer (1985) andEaton and Grossman (1986)—that is, the optimal export policy is a subsidy (tax) if
exporters compete in a Cournot (Bertrand) fashion—holds if the number of domestic
input-suppliers is at an intermediate level. Because a larger number of domestic input
suppliers can enhance welfare by increasing demand for input and exports, the incentives to subsidize become stronger. On the one hand, in the downstream Bertrand case,since the incentive to tax is stronger than that in the Cournot case, it requires a larger
number of input-suppliers than in the Cournot case to realize a positive subsidy. In thedownstream Bertrand case, the threshold number of domestic input suppliers in which
optimal export policy is a subsidy is larger than that in Cournot case. Hence, if thenumber of domestic input-suppliers is at an intermediate level, the optimal policy inthe case of downstream Bertrand (Cournot) is a tax (subsidy).
The rest of the paper is organized as follows. Section 2 presents the model and section 3 derives the equilibrium outcomes in both the downstream Bertrand and Cournot
cases. Section 4 oﬀers conclusions.

2

Model

We consider a third-market model with upstream price competition. The downstream
market consists of two final goods producers, firms H and F ; firm i (= H, F ) is locatedin country i. We call country H (F ) as the Home (Foreign) country. Each firm i exports
its product to the third market.5 The demand and inverse demand functions in the third
5

For simplicity, we omit trade costs in this analysis.

4market are qi =(1−b)a−pi +bpj
1−b2

and pi = a − qi − bqj for i, j = H, F ; i ̸= j, where a > 0,pi , and qi are the price and quantity supplied by firm i, and b (∈ [0, 1)) measures thedegree of product substitutability. Home and Foreign products are perfect substitutes,
where b = 1 and are independently consumed such that b = 0. We assume that firms
have identical production technology, and is linear, where one unit of input is used to
produce one unit of the final good. We also assume that any other production costs arenormalized to zero, that is, the firm’s production cost is the price of a purchased input,
r. To focus on the government’s incentives to choose a policy, we examine the casewhere only the Home country subsidizes.6 Firms’ profits are ΠH ≡ (pH − r + sH )qH
and ΠF ≡ (pF − r)qF , where sH is a per-unit subsidy/tax and sH is a subsidy (tax)when it is positive (negative).
In the upstream world market, there are n (≥ 2) symmetric input suppliers (hereafter called the supplier ). Each supplier k (∈ {1, ..., n}) produces homogeneous inputs
and oﬀers them at a price of rk . We denote supplier k’s and aggregate demand for
inputs by qk and Q (= qH + qF ), respectively. Since firm i purchases inputs from the
supplier oﬀering the lowest price, the demand for supplier k is qk = Q(rmin )/nmin if
the supplier oﬀers the minimum price rk = rmin , where nmin is the number of suppliers
oﬀering the minimum price; the demand is qk = 0 if it does not oﬀer the minimum
price. We assume that the cost for supplier k to produce inputs takes a quadratic formand specify it as (c/2)qk2 , where c (> 0) denotes production eﬃciency. The profit ofsupplier k is πk = qk rk − (c/2)qk2 .

6Ishikawa and Spencer (1999) and Hwang et al. (2007) also consider unilateral intervention.

5

We consider that there are m (∈ [0, n]) suppliers in the Home country and theothers belong to a country other than Home and Foreign countries. The welfare of theHome country is
WH ≡ ΠH + mπk − sH qH .

(1)

The game consists three stages: in the first stage, the Home government decides
the level of sH . In stage two, the input price r is determined through supplier pricecompetition. In the final stage, each firm decides the price (quantity) of its product.
We solve the game by backward induction.
Since we assume that suppliers produce homogeneous inputs with a quadratic cost,
there is a range of Nash equilibria (Dastidar, 1995). Thus, we need to employ somecriterion for selecting equilibrium prices. We use the payoﬀ-dominance criterion. Thisapproach is similar to that in Cabon-Dhersin and Drouhin (2014). Moreover, somestudies focus on a collusive price to narrow the set of Nash equilibria (e.g., Dastidar,
2001 and Gori et al., 2014). This criterion is also similar to our approach. We assume acollusive price higher than or equal to an upper bound of the set of Nash equilibria, while
previous studies with the collusive price criteria make a restriction for the parametersin which the collusive price is in the set of Nash equilibria.
Finally, we explain how to select a price with the payoﬀ dominance criterion in theset of Nash equilibria. We assume symmetric suppliers, so that in any pure strategy
Nash equilibrium in the upstream market, each supplier chooses the same price. We
denote the set of prices each supplier chooses in Nash equilibrium by [r, r̄]. In addition,
we denote a collusive price that maximizes the suppliers’ joint profits by rcol . Because6we assume a range of parameters where r̄ ≤ rcol , the input price each supplier selects
must be equal to r̄ because, r̄ provides the highest profit of suppliers in the set of Nashequilibria.

3

Results

3.1

Downstream Bertrand

We first consider the case where the downstream has Bertrand rivalry. In the third
stage of the game, the FOC to maximize the profit of firm i (= H, F ), ∂Πi /∂pi = 0,
yields its exports qi (r, sH ) and total sales Q(r, sH ).
In the second stage, the input-price r is determined through Dastidar (1995)-type
price competition.7 In pure strategy Nash equilibria, two conditions, πk (n, r, sH ) =[Q(r, sH )/n]r − (c/2)[Q(r, sH )/n]2 ≥ 0 and πk (n, r, sH ) ≥ πk (1, r, sH ) = Q(r, sH )r −(c/2)[Q(r, sH )]2 , must be satisfied: the first is the condition that suppliers do not(2a+sH )c; the second is theraise their prices and yields the lower bound r = 2[(2−b)(1+b)n+c]

condition that suppliers do not reduce their prices and gives the upper bound r̄ =(2a+sH )(1+n)c2[(2−b)(1+b)n+(1+n)c] .

Thus, in equilibrium, the input-price must lie between r and r̄.

Moreover, from the symmetry in suppliers, the collusive price rcol is given by rcol =H)argmaxr πk (n, r, sH ) = [(2−b)(1+b)n+2c](2a+s. These prices yield the following.
4[(2−b)(1+b)n+c]

Lemma 1. (i) rcol > r. (ii) rcol ≥ r̄ if and only if c ≤ cB ≡ [(2 − b)(1 + b)n]/(n − 1).
H)
Proof. (i) Simple algebra yields rcol − r = (2−b)(1+b)n(2a+s4[(2−b)(1+b)n+c] > 0. (ii) Since rcol − r̄ =7

See Dastidar (1995), pp. 27–28. Moreover, Delbono and Lambertini (2016a, 2016b) employ Dastidar (1995)-type price competition. See also Cabon-Dhersin and Drouhin (2014), pp. 432–433, forequilibrium price selection.

7

(2−b)(1+b)n(2a+sH )[−c(n−1)+(2−b)(1+b)n]4[(2−b)(1+b)n+c][(2−b)(1+b)n+(1+n)c] ,≡ cB . Q.E.D.
rcol ≥ r̄ iﬀ c ≤ (2−b)(1+b)nn−1

Lemma 1 states that rcol ∈ (r, r̄) for c > cB . Thus, the second stage input price is acollusive price, r = rcol , for c > cB . However, if r = rcol , the conditions determining thesign of optimal export policy in the downstream Bertrand case (part 2 of Proposition
1) becomes highly complicated. To avoid unnecessary complexity in the analysis andto obtain clear-cut conditions, we assume c ≤ cB . When we put this restriction on c,the second stage input price is r = r̄.
Assumption 1. 0 < c ≤ cB .
On the other hand, even though c > cB , our main result (Proposition 1) does not
qualitatively change. Even if c > cB (i.e., r = rcol ) holds, the optimal policy can
non-monotonically change with respect to product substitutability. Particularly, in acase where n = 2, we obtain a similar result as in part 2 of Proposition 1, even thoughr = rcol holds. We state this result as “Proposition A1” in Appendix A. (See Fig. 5.)
In the first-stage, the Home government chooses sH to maximize (1).8 We use
Kawabata (2010)’s decomposition to express the FOC for welfare maximization as
follows:
(i)

(iii)

(ii)

z (
}|
z }| {
]{
[
( ))
∂r
∂qH
Q
∂WH
∂pH
= (pH − r)
+ qH
+ − qH − m
∂sH
∂sH
∂sH
n
∂sH
(
( ))
m
Q
∂Q
= 0.
+
r−c
n
n
∂sH
{z
}
|
z

}|

{

(2)

(iv)

(2) consists of four eﬀects of a subsidy:9 (i) the horizontal rent-shifting eﬀect on
8
9

The SOC for welfare maximization always holds, i.e., ∂ 2 WH (sH )/∂s2H < 0.
See Kawabata (2010), pp. 119–120.

8

the Home country final product, which corresponds to the first term and is positive;(ii) the terms of trade eﬀect for the Home country final product, which corresponds tothe second term and is negative; (iii) the rent extraction eﬀect from suppliers, whichcorresponds to the third term and is negative if (qH − m(Q/n)) > 0; and (iv) theeﬃciency gain eﬀect from an increase in input production, which corresponds to theforth term and is positive if m > 0.10 (i) and (ii) are the horizontal eﬀects of a subsidy,
whereas (iii) and (iv) are the vertical eﬀects. Terms (iii) and (iv) are both increasing
for the number of domestic suppliers. An increase in m weakens the negative eﬀect(i.e., tax incentive) in the third term, (iii), and strengthens the positive eﬀect (i.e.,c(1+n)(2a+sH )term)= 2[(2−b)(1+b)n+(1+n)c]
subsidy incentive) in the fourth term, (iv): ∂(third
2 > 0 and∂mc(n−1)(2a+sH )∂(fourth term)= 2[(2−b)(1+b)n+(1+n)c]
2 > 0.
∂m

From (2), (A1), and (A2),11 we calculate the optimal export policy assBH =2a(1 − b)(2 + b)n[2(2 + b)cm − (b3 + b2 + c)n − c],D(3)

where D ≡ 2[4(2 − m + 2n) + 4b(1 + n) − b2 (3 + b)(1 − m + n)]cn + (3 + b)c2 (1 + n)2 +8(1 + b)(2 − b2 )n2 > 0. The variables with “B” (“C”) denote the SPNE outcomes in
the case of Bertrand (Cournot) rivalry.
Let us first refer to a result in an existing study, which was obtained by a specificcombination of the number of suppliers in our model. When there is no domestic
supplier and the upstream is monopoly: m = 0 and n = 1; from (3), the optimalexport policy is a tax. This finding matches that of Chou (2011), who finds that even1011(r − (c/n)Q) > 0 holds.
We depict (A1) and (A2) in Appendix B.

9

if there is an upstream monopolist outside the exporter, Eaton and Grossman (1986)’s
result does not change.
Remark 1. (Chou, 2011) If m = 0 and n = 1, sB
H < 0.
If m = 0, the forth term in (2) disappears, and hence, the negative eﬀects of the second
and third terms dominate the positive eﬀects of the first term.
Using (3), we establish the following proposition, the proof of which is given in
Appendix A.
Proposition 1. Suppose that the third-country downstream market has diﬀerentiatedBertrand rivalry and the Assumption 1 holds. Then:
3

2(b +b +c)n+c
B
; sB
1. sB
H = 0 if m = mB ; and sH < 0 if m < mB .
H > 0 if m > mB ≡2(2+b)c

2. If m ∈ (ml , min{mh , n}), then there exist two thresholds, b1 and b2 , such thatBsB
H > 0 for b1 < b < b2 , and sH < 0 for 0 ≤ b < b1 or b2 < b < 1. Here,mh ≡ mB | b=1 = [n(3 + c)]/6c and ml ≡ mB | b=bl , where bl (∈ (0, 1)) is aminimizer of mB .

Proposition 1 oﬀers two important assertions. The first is that there is a thresholdin the number of domestic suppliers that makes the optimal policy a subsidy. The
second one demonstrates that the optimal policy can non-monotonically change, thatis, tax–subsidy–tax, as the degree of substitutability between final goods becomes large.
Fig. 1 illustrates the second part of Proposition 1. (See also Panel (b) in Fig. 2.)
We explain the first assertion as follows. For a given number of suppliers n, an
increase in the number of domestic suppliers m strengthens the positive eﬀect of the10

forth term in (2), whereas it can weaken the (negative) eﬀect of the third term. This
makes the motive to subsidize stronger.12[Figure 1 around here][Figure 2 around here]
To explain the logic behind the second assertion, for given c and n, let us consider
three diﬀerent sizes of m. Panels (a)–(c) in Fig. 2 illustrate the optimal export policy
corresponding to each size of m. When m is small, the third term in (2) (tax incentive)
is large and the fourth term (subsidy incentive) is small. Then, the optimal policy is atax (see Panel (a), in the “m = 3” case). In contrast, when m is large, the magnitudes inthe third term (tax incentive) is small and those in the fourth term (subsidy incentive)
is large. Then, the Home government considers domestic suppliers to be important andoﬀers a subsidy to firm H to promote exports and domestic input production (see Panel(c), in the “m = 7” case). The limit case where b = 1 implies homogeneous Bertrand
competition in the downstream market, and hence, the firm’s rent vanishes. Thus, theoptimal policy approaches 0 as b approaches 1, regardless of the size of m.
When m is intermediate, the role of b becomes more significant. In diﬀerentiatedBertrand rivalry, it is well-known that firms’ output is U-shaped for b.13 From thisfact, firm H’s exports tend to increase as b goes over a certain level. The positive eﬀectof the fourth term depends on total sales (outputs), so it is also U-shaped for b. On
the one hand, since the positive eﬀect of the first term depends on the prices of boththe input and the final product, it is not necessarily U-shaped for b. In contrast to the
12
13

∂sB
H /∂m > 0. See Appendix D.
Substituting sH = 0 into qH of (A1), we can immediately find this characteristic.

11

positive eﬀect of the fourth term, the negative eﬀects of the second and third terms
can be inverted-U shaped for b. That is, if b aﬀects the U-shape of the outputs of firm
H, the tax-incentive decreases as b increases when it is below a certain level because
the domestic firm’s exports decrease. However, the incentive for taxation increases asb goes above a certain level because the domestic firm’s exports increase in this case.
A rise in m reduces the magnitude of the third term. On the one hand, if m is notlarge relative to n, the magnitude of the third term is not necessarily small because thevertical rent-extraction can derive from n−m non-domestic suppliers. This implies that
under a certain size of m, a change in the negative eﬀects (second and third terms) due
to an increase in b can dominate a change in the positive eﬀects (first and fourth terms).
When m is intermediate, a reduced tax dominates if b is a smaller value. Within sucha range of b, the optimal policy can change from a tax to subsidy as b increases. If b
goes above a certain level and it enters the dominant area of “an increase in the tax
incentive,” the optimal policy can change from a subsidy to a tax as b increases (seePanel (b), in the “m = 5” case).
In particular, the second part of Proposition 1 has a significant policy implication
because the non-monotonicity in the optimal export policy (tax–subsidy–tax) implies
that a big mistake does not matter, but a small mistake can be fatal. The optimal taxes
can appear from two regions of a lower b and a higher b. Thus, even if practitioners
incorrectly recognize that “b is high” when its actual value is low, because they choose
an export tax, the welfare loss may not be so large. In contrast, if there is a small
gap between the practitioner’s recognition of b and its actual value, he or she may

12unfortunately adopt the policy that is not recommended. This possibly yields a serious
welfare-loss.

3.2

Downstream Cournot

We next consider the case where the downstream has Cournot rivalry. To distinguishbetween the Bertrand and Cournot cases, we mark the variables in the Cournot casewith a star (“∗”). In the third stage of the game, the FOC for the profit maximizationof firm i (= H, F ), ∂Πi /∂qi = 0, yields its exports qi∗ (r, sH ) and total sales Q∗ (r, sH ).
In the second-stage, the input-price r∗ is determined in a similar manner as in theprevious section and it yields the second-stage outcomes (see Appendix B, (A3)).
From (2), (A3), and (A4),14 the optimal export policy in the Cournot case issC
H =2a(2 − b)n 2[b n − (1 − b)(1 + n)c + 2(2 − b)cm],E(4)

where E ≡ 2 [((1 − b)(5 + b) + 3)(1 + n) + ((4 − b)b − 4)m] cn + (3 − 2b)c2 (1 + n)2 +8(2 − b2 )n2 > 0. Here, we assume that m < min{m0 , n}. (For m0 , see Appendix C.)
Using (4), we obtain the following proposition, the proof of which is given in Appendix A.
Proposition 2. Suppose that the third country downstream market is a diﬀerentiated
Cournot rivalry. Then: (I) (i) If b > eb, or (ii) m > mC and b < eb, then sC
H > 0. (II)Ce
If m = mC and b < eb, then sC
H = 0. (III) If m < mC and b < b, then sH < 0. Here,√2neb ≡ −(1+n)c+ c(1+n)[c(1+n)+4n] , and 0 < eb < 1.
,mC ≡ (1−b)(1+n)c−b2n2(2−b)c[Figure 3 around here]
14

Appendix B illustrates (A3) and (A4).

13

Fig. 3 illustrates Proposition 2 in the b-m plane. We start by examining the casewhere m = 0, in which the fourth term in (2) disappears and the vertical eﬀect isequal to the “vertical rent-extraction eﬀect” (tax incentive). When m = 0, the export
policy depends on b. A smaller b corresponds to weaker competition between firms, andthus, the horizontal rent-shifting eﬀect (i.e., the positive eﬀect of the first term in (2))
is weak. In such case, the vertical rent-extraction eﬀect is dominant, so the optimal
policy is a tax. Conversely, if b is close to unity, competition between firms is keener
and the horizontal rent-shifting eﬀect is stronger. Then, the optimal policy becomes a
subsidy. This result represents Chou’s (2011) argument.
Remark 2. (Chou, 2011) Suppose m = 0 and n = 1. If b < (≥) −c +√c(2 + c),then sC
H < (≥) 0.
Since an increase in m strengthens the positive eﬀect of the fourth term and weakens
the negative eﬀect of the third term, the tax incentives can become weaker and subsidy
incentives can become stronger as m increases. Therefore, when the size in m is large
relative to n, the optimal policy becomes a subsidy (see Fig. 3). This corresponds to
Kawabata’s (2010) result, which is compatible with a specific combination of m = 1
and n = 2 in our model.
Remark 3. (Kawabata, 2010) If m = 1 and n = 2, sC
H > 0.
Combining propositions 1 and 2, we establish the following proposition, the proofof which is given in Appendix A.

14

C
Proposition 3. (I) Suppose b > eb. (i) If m < mB , then sB
H < 0 and sH > 0; (ii) ifCBem > mB , then sB
H > 0 and sH > 0. (II) Suppose b < b. (i) If m < mC , then sH < 0
BCand sCH < 0; (ii) if mC < m < mB , then sH < 0 and sH > 0; (iii) if m > mB , thenCsB
H > 0 and sH > 0.

Proposition 3 shows the condition for which the conventional results hold (parts (i)in (I) and (ii) in (II)): when the number of domestic suppliers is intermediate, downstream Bertrand rivalry yields a tax and Cournot rivalry yields a subsidy in verticalstructure with upstream price competition (see also Fig. 3).
We find this result because in the Bertrand case, the threshold value of m thatmakes the optimal export policy a subsidy is larger than that in the Cournot case (i.e.,mC < mB ). In the case of downstream Bertrand rivalry, if there is no upstream sector,
the negative eﬀect of the second term in (2) (tax incentive) dominates the positiveeﬀect of the first term (subsidy incentive), and thus the optimal export policy is a tax(Eaton and Grossman, 1986). Additionally, if there is an upstream market and m = 0
holds, because the vertical rent-extraction eﬀect in the third term points to a tax, theoptimal policy is a tax for any level of product substitutability (Chou, 2011). In bothcases of downstream Bertrand and Cournot competition, an increase in m weakens thenegative eﬀect of the third term (tax incentive) and strengthens the positive eﬀect of
the fourth term (subsidy incentive), but the tax incentives in the Bertrand case arestronger than that in the Cournot case. Therefore, mC < mB holds.
Welfare comparison. For a large m, the Home country welfare in the case of downstream
Bertrand competition can be higher than that in the case of downstream Cournot
15

competition. Fig. 4 shows this in the b-m plane. We examine this result here.
[Figure 4 around here]
Let us first focus on the fact that a larger m improves the Home country’s welfare, regardless of the competition mode in the downstream market. (∂WHl /∂m > 0,l = B, C. See Appendix D.) Since the incentives to subsidize become stronger as m increases, slH increases with m (∂slH /∂m > 0). A higher subsidy (lower tax) shifts input
demand upward. This demand expansion allows suppliers to set a higher price, so an
increase in m raises the input price (∂rl /∂m > 0). Additionally, a higher subsidy raises
total sales (or exports) because it increases the domestic firm’s exports more than it
reduces the foreign firm’s exports (∂Ql /∂m > 0). Hence, an increase in m promotes thedomestic firm’s exports, magnifies input demand, raises the input price, and improves
welfare.
On the one hand, the competition is tougher in Bertrand rivalry rather than in
Cournot rivalry, so the eﬀects of export promotion (restriction) tend to work stronger
in the Bertrand case. When m is small enough, the export policy is always a tax inthe Bertrand case. Then, since the suppliers’ part of the profit (i.e., mπk ) is small andthe export activity of firm H is dampened, the welfare level in the Bertrand case canbe lower than that in the Cournot case. In contrast, when m is large enough and theexport policy is a subsidy in the Bertrand case, because the suppliers’ part of the profit
is large and the eﬀects of export promotion are also stronger, the welfare level can behigher than that in the Cournot case15 (see Fig. 4).
15

However, when b is suﬃciently close to 1, since the firms’ profit is close to 0 in the Bertrand case,the welfare level in the Cournot case exceeds that in the Bertrand case.

16

4Conclusion

This paper incorporates Dastidar (1995)-type price competition in the intermediate
input into a standard export rivalry model with a vertical structure and studies thenature of the optimal export policy. Although input-suppliers have a quadratic cost
and the equilibrium input price has a certain range, by adopting a similar approach as
Cabon-Dhersin and Drouhin (2014) (i.e., payoﬀ-dominance criterion) do, we narrow therange of input prices to a single equilibrium price and consider diﬀerentiated Bertrand
and Cournot competition in the downstream third market.
We first show that in the case of downstream Bertrand competition, the optimalexport policy can be tax–subsidy–tax, corresponding to the degree of product substitutability. This non-monotonicity in the export policy gives the following policy implication: there is a case in which a large mistake can be permissible, but a small mistakecan be impermissible. The optimal export policy becomes a tax when the degree ofsubstitutability is both low and high. Hence, when the degree of substitutability islow, even if the practitioner accidentally recognizes that its level is high, the realized
policy be an export tax and the optimal and realized policies are consistent. Therefore,a large mistake is permissible. However, if the practitioner’s recognition of productsubstitutability is slightly higher than that of the actual level, the realized policy be
an export subsidy, which is the opposite to the optimal policy. Hence, a small mistake
may be impermissible.
We also demonstrate that if the number of domestic input suppliers is intermediate,
the conventional results, that is, the optimal export policy is a tax (subsidy) when17

exporters compete in a Bertrand (Cournot) fashion, holds. A larger number of domestic
input suppliers strengthens subsidy incentives, so in both the Bertrand and Cournot
cases, the optimal export policy becomes a subsidy when the number of domestic
suppliers is large enough. Because the tax incentive in Bertrand competition is stronger
than that in Cournot competition, to switch the optimal export policy from a tax to
a subsidy in the case of downstream Bertrand competition, it is necessary to have alarger number of domestic suppliers compared to the Cournot case.
In this paper, we considered upstream price competition in a standard third-market
export-rivalry model. On the one hand, it may be possible to extend our model to a
two-way trade environment. In such a situation, examining the role of upstream pricecompetition may be an interesting consideration. However, this argument is beyond
the scope of our analysis and is left for future work.

Appendices
Appendix A. Proofs and Proposition A1.
3
2
Proof of Proposition 1. From (3), solving sB
H = 0 for m, we have mB = [nb + nb +c(1 + n)]/[2(2 + b)c]. Then, we obtain the first assertion.
Next, we consider the second assertion. Diﬀerentiating mB wrt b, we get
2nb3 + 7nb2 + 4nb − c(1 + n)
∂mB
=
.
∂b
2c(2 + b)2
To prove the non-monotonicity of sB
H , we show that mB is a convex function and its
first derivative wrt b takes a negative value at b = 0 and a positive value at b = 1. First,

18

we have ∂ 2 mB /∂b2 = [(4 + 12b + 6b2 + b3 )n + c(1 + n)]/[(2 + b)3 c] > 0. Second, we have∂mB /∂b|b=0 = −c(1+n)/(8c) < 0. Finally, we have ∂mB /∂b|b=1 = (13n−cn−c)/(18c),
which is a decreasing function for c. Since we assume c ≤ cB , ∂mB /∂b|b=1 takes
the minimum value at c = cB . Substituting c = cB , we have ∂mB /∂b|b=1, c=cB =(11n − 15 − b − bn + b2 + b2 n)/[18(2 − b)(1 + b)] > 0. The inequality is satisfied since
we assume n ≥ 2. Hence, ∂mB /∂b|b=1 > 0.
From these and the continuity of mB , there exists a unique bl ≡ argminb mB in(0, 1) such that for any b < bl , mB decreases with b, and for any b > bl , mB increases
with b.
Here, we define mh = min{mB |b=0 , mB |b=1 }, where mB |b=0 > mB |b=1 if 4n/(1 +n) < c < cB ; mB |b=0 ≤ mB |b=1 if 0 < c ≤ 4n/(1 + n). Since n − mB |b=0 = (3n − 1)/4 >0, we have n > mh .
From the discussion above, we have some m′ ∈ (ml , min{mh , n}) such that sB
H >0
for b1 < b < b2 , and sB
H < 0 for 0 ≤ b < b1 or b2 < b < 1. Q.E.D.
Proof of Proposition 2. From (4), solving sC
H ≤ (≥) 0 wrt m, we have m ≤ (≥)2n(4−b)bn+(1+n)c∂mC
mC ≡ (1−b)(1+n)c−b. mC |b=0 = 1+n< 0, and mC |b=1 =4 > 0, ∂b = −2(2−b)c2(2−b)2 cn− 2c< 0, so mC = 0 at some b ∈ (0, 1). Solving mC ≥ 0 wrt b, we obtain b ≤√√eb ≡ −(1+n)c+ c(1+n)[c(1+n)+4n] > 0. Since 1 − eb = (c+2n+cn)− c(1+n)[c(1+n)+4n] and2n2n(c + 2n + cn)2 −(√)2c(1 + n)[c(1 + n) + 4n] = 4n2 , 0 < eb < 1. Hence, if b > eb then

C
e C
sC
H > 0. When b < b, sH < 0 if m < mC , and sH > 0 if m > mC . Q.E.D.
2

2)n+(1+n)c]> 0.
Proof of Proposition 3. From mB > 0, mB − mC = b [(4+2b−b2(2−b)(2+b)c+b )n+(6−b−3b )(1+n)c
Further, we obtain m0 − mB = (16−10b −3b2(2−b)(2+b)c> 0. From these and
2

3

19

4

2

Propositions 1 and 2, we obtain Proposition 3. Q.E.D.

The collusive input price case: r = rcol (i.e., c > cB ).
Proposition A1. Suppose c > cB and 2 = n > mcol
≡ minb mcol , where mcol ≡l
2+b+b3 +c2(2+b)(1+b)(2+b) . Then:colcol
1. scol
H > 0 if m > mcol ; sH = 0 if m = mcol ; and sH < 0 if m < mcol .
col
col
col
col
col
2. For m′′ ∈ (mcoll , min{mh , 2}), we have sH > 0 for b1 < b < b2 , and sH < 0 forcolcol
0 ≤ b < bcol
1 or b2 < b < 1, where mh ≡ mcol |b=1 .

Proof. For simplicity, we assume n = 2. From Lemma 1, we have rcol < r̄ if c > cB .
We assume the range of c. Then, the equilibrium input price is r = rcol . Substituting
it into WH and solving the first-order condition ∂WH /∂sH = 0 for sH , we obtain the
optimal export policy as
scol
H =

2a(1 − b)(2 + b)(2 + b + c − 8m − 8bm + 2b2 m + b3 (1 + 2m)),
Dcol

where Dcol ≡ −60 − 28c − 3c2 + 16m − b(64 + 18c + c2 − 8m) + b2 (33 + 10c − 20m) +b3 (35 + 4c − 10m) + b4 (4m − 5) + b5 (2m − 3).
Solving scol
H = 0 for m, we havem=

2 + c + b + b3
≡ mcol .
2(2 + b)(2 + b − b2 )Then, we obtain the first assertion.
To prove the second assertion, we show that mcol is a convex function and its first
derivative for b takes a negative value at b = 0 and a positive value at b = 1.

20

First, we have
∂ 2 mcol24 + 60b + 54b2 + 41b3 + 39b4 + 15b5 − b6 + (20 − 9b2 + 8b3 + 6b4 )c=.
∂b2(4 + 4b − b2 − b3 )3This function increases with c. Since we assume c > cB , to calculate the minimum
value of this function, we evaluate it at c = cB . Then, we have
∂ 2 mcol104 − 4b − 18b2 + 73b3 + 24b4 − 13b5=
> 0.
∂b2 c=cB(1 + b)2 (4 − b2 )3Hence, ∂ 2 mcol /∂b2 > 0 always holds.
Next, we evaluate ∂mcol /∂b at b = 0 and b = 1. Then, we have ∂mcol /∂b|b=0 =−(1 + c)/8 < 0 and ∂mcol /∂b|b=1 = (28 + c)/72 > 0. Moreover, we have ∂mcol /∂b|b=0 −∂mcol /∂b|b=1 = (c − 2)/24, which increases with c. Hence, to evaluate it at c =cB , we obtain the minimum value (1 + b − b2 )/12. Then, we obtain ∂mcol /∂b|b=0 >∂mcol /∂b|b=1 . From these results and the continuity of mcol , there exists a uniquebcol ≡ argminb mcol in (0, 1) such that for any b < bcol , mcol decreases with b, and forany b > bcol , mcol increases with b.
Here, let mcol
≡ minb mcol and mcollh = mcol |b=1 . From the discussion above, if
col
col
col
mcol
< 2 (= n), we have some m′′ ∈ (mcolll , min{mh , 2}) such that sH > 0 for b1 <
col
col
col
b < bcol2 , and sH < 0 for 0 ≤ b < b1 or b2 < b < 1.

Finally, we show that at c = 44/5, three types of optimal policies occur: tax at
m = 0, tax–subsidy–tax at m = 1, and subsidy at m = 2. Q.E.D.
[Figure 5 around here]

21

Appendix B. Second-stage outcomes
The second-stage outcomes in the downstream Bertrand case are:



pH


2 )n+(1+n)c]−[4(1+b)n+(1+n)c]s(2a+sH )(1+n)cH; r = 2[(2−b)(1+b)n+(1+n)c], = 2a(2+b)[(1−b2(2+b)[(2−b)(1+b)+(1+n)c]qH2 )+(1+n)c]s(2a+sH )nH= 2a(1−b)(2+b)n+[2n(2−b2(1−b)(2+b)[(2−b)(1+b)n+(1+n)c] ; Q = (2−b)(1+b)n+(1+n)c .





(A1)

Comparative statics result for (A1):∂pH∂sH−[4(1+b)n+(1+n)c]= 2(2+b)[(2−b)(1+b)+(1+n)c]
< 0;

∂qH
∂sH

2n(2−b2 )+(1+n)c= 2(1−b)(2+b)[(2−b)(1+b)n+(1+n)c]
> 0;






(1+n)c∂r∂sH = 2[(2−b)(1+b)n+(1+n)c] > 0,




∂Q
n

∂sH = (2−b)(1+b)n+(1+n)c > 0.

(A2)

The second-stage outcomes in the Cournot case are:2a(2−b)[(1+n)c+n]−[2(2−b2 )+(1−b)(1+n)c]s
(2a+sH )(1+n)cH; r∗ = 2[(2+b)n+(1+n)c]
, 

p∗H

=

∗
qH

(2a+sH )nH∗= 2a(2−b)n+[(1+n)c+4n]s2(2−b)[(2+b)n+(1+n)c] ; Q = (2+b)n+(1+n)c .

2(2−b)[(2+b)n+(1+n)c]





(A3)

Comparative statics result for (A3):


∂p∗H
∂sH

2 )+(1−b)(1+n)c]= −[2(2−b2(2−b)[(2+b)n+(1+n)c] < 0;

∗
∂qH
∂sH

(1+n)c+4n= 2(2−b)[(2+b)n+(1+n)c]> 0;
(1+n)c∂sH = 2[(2+b)n+(1+n)c] > 0,
∂r∗

∂Q∗
n
∂sH = (2+b)n+(1+n)c > 0.





(A4)

Appendix C. SPNE outcomes
Bertrand case:a(2 + b)n[2(2 − b2 )n + c(1 + 2m + n)],
D
an
qFB =
[2(4 + 2b − b2 )n + c(4(1 − m + n) + b(1 − 2m + n))],Dac(1+n)a2 nrB =[(8+4b−3b2 −b3 )n+c(3+b)(1+n)]; WHB =[(1−b)(2+b)2 n+2(3+b)cm].
D
D

B
qH
=

( B )2
2
The equilibrium profit of firm i is ΠB, i = H, F . The supplier k’s
i = (1 − b ) qi
( B )22n
profit is πkB = c(1+n), k ∈ {1, . . . , n}.
2 r

22

Cournot case:a(2 − b)n[4n + (1 + 2m + n)c]
,
E
an
qFC =
[2(4 − 2b − b2 )n + (4 − 3b)(1 + n)c − 2(2 − b)cm],Ea(1 + n)ca2 n[2(3 − 2b)cm + (2 − b)2 n]rC =[(8−4b−b2 )n + (3−2b)(1+n)c]; WHC =
.
E
E

C
qH
=

The equilibrium profit of firm i is ΠCi =2nc(1+n)2( C )2
qi . The supplier k’s profit is πkC =( C )2r. Here, to ensure a positive quantity, we assume
m < m0 ≡2(4 − 2b − b2 )n + (4 − 3b)(1 + n)c(> 0),
2(2 − b)c

which is equivalent to qFC > 0.

Appendix D. Comparative statics result for SPNE outcomes
From the derivative of the outcomes, we get∂sB
4a(1 − b)(2 + b)2 cn(2 − b)(1 + b)(8 + 4b − 3b2 − b3 )n2
H
=
∂m
D2
4a(1−b)(2+b)2 cn[(14+9b−5b2 −2b3 )n(1+n)c+(3+b)(1+n)2 c2 ]
> 0,
+
D2
∂sC
4a(2−b)2 cn[(2+b)(8−4b−b2 )n2 +(14−5b−3b2 )n(1+n)c+(3−2b)(1+n)2 c2 ]
H
=
> 0,
∂m
E2
∂rB
2a(1 − b)(2 + b)2 c2 n(1 + n)[(8 + 4b − 3b2 − b3 )n + (3 + b)c(1 + n)]
=
> 0,
∂m
D2
2a(2 − b)2 c2 n(1 + n)[(8 − 4b − b2 )n + (3 − 2b)c(1 + n)]
∂rC
=
> 0,
∂m
E2
∂QB
4a(1 − b)(2 + b)2 cn2 [(8 + 4b − 3b2 − b3 )n + (3 + b)c(1 + n)]
=
> 0,
∂m
D2
4a(2 − b)2 cn2 [(8 − 4b − b2 )n + (3 − 2b)c(n + 1)]
∂QC
=
> 0,
∂m
E2

23

∂WHB
2a2 cn[(8 + 4b − 3b2 − b3 )n + (3 + b)c(1 + n)]2=
> 0, and∂m
D2∂WHC
2a2 cn[(8 − 4b − b2 )n + (3 − 2b)c(1 + n)]2
=
> 0.
∂m
E2

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26

m

mB

B
sH
>0

mh
m′
ml
B
sH
<0

0

b2

b1

1

b

Figure 1: The area of the non-monotonic export policy under c ≤ cB .

27

0.2

0.4

0.6

0.8

b
1.0

0.006
0.004

-0.01

0.002

-0.02
0.2

0.4

0.6

0.8

b
1.0

-0.03
-0.002-0.04-0.004-0.05-0.006-0.06-0.008Panel (a): m = 3 (tax).

Panel (b): m = 5 (tax–subsidy–tax).

0.05
0.04
0.03
0.02
0.01

0.0

0.2

0.4

0.6

0.8

b
1.0

Panel (c): m = 7 (subsidy).
Figure 2: Graph of sB
H /a for b. (c = 1.9 and n = 20.)

28

m
sB
H >0

mB

sB
H <0
sCH > 0sCH < 0

mC

0

eb

b
1

Figure 3: The area of “conventional results.”

20

m

15

WBH > WCH

10

5

0
0.0

WBH < WCH
0.2

0.4

0.6

0.8

1.0

b
Figure 4: Welfare comparison. (a = 1, c = 1.9, and n = 20.)

29

m
mcol
scol
H >0
mcol
h
m′′
mcol
l
scol
H <0
0

bcol
2

bcol
1

1

b

Figure 5: The area of non-monotonic export policy under c > cB .

30