Strategic Trade Policy under Isoelastic
Demand and Asymmetric Production Costs
Akio Matsumoto1
Department of Economics
Chuo University

Nobuko Serizawa2
Department of Economics
Niigata University

June 11, 2004

1 Correspongind author:

742-1, Higashi-Nakano, Hachioji, Tokyo, 192-0393,
Japan. akiom@tamacc.chuo-u.ac.jp; +81-426-74-3351(tel); +81-426-74-3425 (fax).
2 8050, 2-no-cho, Ikarashi, Niigata, 950-2181, Japan. serizawa@econ.niigatau.ac.jp; +81-25-262-6568 (tel/fax).

Abstract
We demonstrate that whether a good of a rival firm is a strategic substitute or
astrategic complement is endogenously determined when the market inverse
demand is hyperbolic. The relative competitiveness, which is expressed by
the ratio of firms’ marginal costs, is the key determinant. We derive optimal
trade policies, which are dependant upon the firms’ form of strategic action.
In particular, the optimal policy recommendation for the home governmentis to give an export subsidy to its firm if a choice variable of the foreign
firm is a strategic substitute and to levy an export tax if it is a strategic
complement.

Acknowledgement
Preliminary version of this paper was presented at the 43rd Annual Meeting of the Western Regional Science Association held in Maui, Hawaii, February 25-28, 2004 and an workshop on international trade held in Niigata,Japan, May 21, 2004. The authors wish to thank particiants, Kenneth E.
Corey and Jota Ishikawa for comments. Akio Matsumoto is grateful for financial supports from Chuo University (Joint Research Grant, 0382) andthe Japan Ministry of Education, Culture, Sports, Science and Technology(Grand-in-Aid for Scientific Research (B), 15330037).

1

Introduction

The market is imperfectly competitive, either because the number of firms
in that market is few, because the goods are diﬀerentiated, or because there
is some kind of economies of scale. In reality, we see many imperfectly competitive industries, in which firms compete fiercely both domestically and internationally. In such an imperfectly competitive international market, governments may be motivated to introduce trade policies that includes tariﬀs,
export subsidies and taxes, even if such policy intervention is not justified.
If governments can aﬀect, or more precisely, strategically alter market structure by introducing policies, how would behavior of the firms be impacted
or what would the impacts on the market be. To address these issues andinvestigate their motive, researchers have contributed contributed numerous
trade theory papers since 1980s. In many international oligopoly models, it
has been shown that a government can introduce trade policy so as to increase its domestic welfare by shifting monopoly rent from foreign rival firms
to its own firms. This is why such government policy is called strategic trade
policy. But, this does not mean that there is a clear understanding to fill thegap between the normative and positive implications of economic activities
among firms and governments and between firms and governments.
OF the vast amount of literature on strategic trade policy in imperfectlycompetitive markets, the pioneering Brander and Spencer [1985] show thatwhen the home government intends to shift monopolistic rent from foreign todomestic firms an export subsidy is optimal in a model, in which one homefirm and one foreign firm of constant-return-to-scale technology produce homogeneous goods and compete in quantities in a third market. An increase in
the domestic export subsidy raises the domestic firm’s output (market share)and its profit. Eaton and Grossman [1986], however, derive an opposite resultin that an export tax for the home firm, which raises profits of the foreign
firm, is optimal when two firms compete in prices. What is the source of thissharp contrast? Brander and Spencer [1985] assume that a firm’s own marginal revenue declines with an increase in the output of the other firm; thatis, players’ strategies are said to be strategic substitutes.1 This assumption isreversed in Eaton and Grossman [1986] where that firm’s own marginal revenue increases with an increase in the price of the other firm; that is, players’
strategies are strategic complements. It is the diﬀerence in the assumption
on firms’ strategic behavior, which is reflected in the opposite tangencies offirms’ reaction function in the output plane, that plays the central role in1

For strategic substitutes and strategic compliments, see Bulow, Geanakoplos andKlemperer (1985). Geometrically, players’ strategies are said to be strategic substitutes(compliments), if their best response functions are downward (upward) sloping.

1

driving opposite policy implications. It is well known that this critical assumption relates to the model’s stability conditions and the curvature of theinverse demand functions as well as the elasticity of the demand curve. (Seefor example, Brander and Spencer [1984a] and [1984b] and Jones [1987].) Focusing on these issues, Bandyopadhyay [1997] simplifies Brander and Spencer[1985] and shows that demand elasticities and cost asymmetry determine optimal trade policy direction - either export subsidy, export tax, or free trade- when one government unilaterally intervenes. Bandyopadhyay [1997] also
demonstrates that, depending on demand elasticities, the Nash equilibriumin the policy game may be unstable when both governments bilaterally intervene. However, Bandyopadhyay [1997] does not explain the mechanism
whereby the diﬀerence in marginal costs aﬀects firms’ rivalry in the goodmarket in relation to demand elasticities and the governments’ interventions.
In this paper, we extend Brander and Spencer [1985] by introducing a hyperbolic inverse demand curve which is the second familiar demand functionin economics. By specifying a unit elastic demand, we examine how sucha minor change alters government and firm behavior as well as the characteristics of the equilibria and clarify how the cost asymmetry relates to theunilateral trade policy. We show that: (i) the diﬀerence of constant marginal
costs endogenously determines whether firms’ strategies are strategic substitutes or complements when the inverse demand function is hyperbolic, andwe clarify conditions for this; and (ii) trade induced by export subsidies of
the home government is not necessarily beneficial for the home firm.
The study is set out as follows. In Section 2, we model a simple nonlinear
international duopoly model, and derive some specific features of firms’ rivalry in the market. There is no government intervention in the market. Thisis altered in the next Section. It is demonstrated that reaction functions aremound-shaped and that whether goods are strategic substitutes or strategic
complements is endogenously defined. Section 3 considers strategic trade
policies in a Stackelberg game, in which the home government moves first
before the international Cournot-Nash duopolistic competition starts. We
derive the Stackelberg equilibrium and the optimal level of the trade policy.
Then we demonstrate that the home government’s trade policy, for example
a production subsidy, can provide a strategic advantage to the home firm if
a choice variable of the foreign firm is a strategic substitute and a strategic
disadvantage if a choice variable of the foreign firm is a strategic complement.
Finally, concluding remarks are provided in Section 4.

2

2International Duopoly with Hyperbolic Demand Function

2.1

Cournot-Nash Competition

We consider an international duopoly model, in which the home firm produces output x with constant marginal cost cx , and the foreign firm producing
output y with constant marginal cost cy . Goods are homogeneous, and weassume that firms export their products only to the third country and compete in quantities in Cournot fashion with price p. For simplicity, we assume
that neither firm faces any domestic demands in their domestic markets. We
further assume the inverse demand function to be hyperbolic,2p(x, y) =

1
.
x+y

(1)

The profits of the home firm and foreign firm are, therefore,Πh (x, y) = p(x, y)x − cx x,(2)

Πf (x, y) = p(x, y)y − cy y.
Each firm maximizes its profit with respect to its own output given therival’s output. Equating the partial derivatives of profit functions to zeroand arranging terms yieldsry− y,rh (y) ≡cx(3)rx
rf (x) ≡− x,
cywhere rh (y) is a reaction function of the home firm, and rf (x) is a reaction
function of the foreign firm. Each reaction curve is mound-shaped with its
highest point for x and y such that x = y. Substituting y = x into the firstequation of (3) and solving for x gives the maximum value of the home firm’s
output, denoted by xM , and the associated maximizer of y, denoted by yM ,xM =1and yM = xM .
4cx

(4)

1
This function is a special case of p(x, y) = (x+y)λ where λ is a reciprocal of market
demand elasticity. The general case is considered in Matsumoto and Serizawa [2003].
2

3

Solving rh (y) = 0 for y gives two levels of outputs, the lower yl and the higher
yh , that lead to zero optimal output of the home firm,
yl = 0 and yh =

1
.
cx

(5)

Since the foreign firm is exactly the same as the home firm, its reaction
function has the same properties where x, y, cx and the subscript, h, for thehome firm are, respectively, replaced with y, x, cy and the subscript, f , forthe foreign firm. To avoid the notational confusion, we denote the maximum
and the maximizer of the reaction curve of the foreign firm by ym and xm .
It can be checked that
drf (x)T 0 according to x S xm
dx

(6)

where the maximum output of the foreign firm and the associated maximizer
are
1
ym =
and xm = ym ,(7)4cy
and the home firm’s outputs that leads to zero optimal output of the foreign
firm are1
xl = 0 and xh = .
(8)cyAbove all, the mound-shaped reaction curves of the home and the foreign
firms start at origin, which have the peaked points of (yM , xM ) and (xm , ym )respectively, through which the critical line y = x or x = y passes, and thendrops to zero at y = yh on the y-axes and x = xh on the x-axes as illustrated
in Figure 1(A) and Figure 1(B).3
Proposition 1 The reaction curves of duopolistic firms are mound-shaped
when the inverse demand function is hyperbolic.
3

Note that the x = y or y = x locus is the diagonal. The ratio between the x-axis andthe y-axis is appropriately adjusted. Please ignore the dashed lines and k for now.

4

x

y

x=y

1
x= ÅÅÅÅÅ y
k

xM
xc

yc

yc

yh

y=kx

ym

C

yM

y= x

y

C

xm

(A) Home Firm with k < 1

xc

xh

x

(B) Foreign Firm with k > 1

Figure 1. Mound-shaped Reaction Curves

2.2

Strategic Substitutes and Complements

The mound-shaped reaction curve shows a sharp contrast to the traditional
linear reaction curve, either downward-sloping or upward-sloping, and plays
the central role in the following analysis. Before proceeding, we digress todetect a possible economic source of these curves. However, we focus only on
the home firm because the same source is shared by the foreign firm which
is symmetric with the home firm.
Since the cost function is assumed to be linear, a possible source of thenonlinearity should be found in the revenue function. Revenue is the pricetimes the quantity sold, and the marginal revenue of the home firm has thewell-known formula,∂p
MR = p + x .
(9)∂x(9) implies that a change of its own output generates two types of eﬀects onrevenue. The first term p implies changes of revenue caused by a unit changeof output, keeping the price level constant, while the second term describeschanges of revenue caused by a unit change of price, keeping the output level
constant. An alternative expression of (9) is also familiar
MR = p(1 −

5

1
),
|εx |

(10)

where εx is the price elasticity of the home firm’s demand given output of
the foreign firm and can be expressed as¯¯¯ d log x ¯ x + y¯¯(11)¯ d log p ¯ = x .

(10) and (11) means that if y is positive, the elasticity |εx | is greater thanunity, and thus the marginal revenue of the home firm is positive.
We are now concerned with how the home firm’s marginal revenue changes
in response to changes in the foreign firm’s choice of output. When outputof the foreign firm increases, there are two sorts of eﬀects according to thedecomposition of the marginal revenue given in (9): one aﬀects the first term
p via shifting the demand curve and the other aﬀects the second term throughchanging the slope of the demand curve. The first eﬀect is to decrease themarginal revenue as the increase in output makes the price decrease in order
to sell the larger quantity, so revenue of the home firm decreases, provided
that the level of its output is constant. The second eﬀect is to increase themarginal revenue by changing the slope of the demand curve of the home firm.
The opposite direction of these two eﬀects implies that the total change canbe of either, depending on the relative magnitude between the increases andthe decreases. To see the total eﬀect under the hyperbolic inverse demandin (1), we diﬀerentiate MR with respect y to get∂MRx(2 − |εx |) .
=∂y(x + y)2(12)

Thus the marginal revenue increases when y increases if the price elasticityof the home firm’s demand is less than two in absolute value. Similarly, themarginal revenue decreases when y increases if the price elasticity is greaterthan two in absolute value. Returning to the profit function of the home firmand deriving its cross derivative we have
µ
¶
∂ ∂Πh
∂MR
=
.
(13)
∂y ∂x
∂y
Due to the definition given by Bulow, et al [1985], output of the home firm canbe a strategic substitute or a strategic complement according to the marginal
profit being negative or positive in response to a change in the foreign firm’s
output. Thus (12) and (13) implies the following.
Proposition 2 Under the hyperbolic demand, the home firm regards its output as a strategic complement when demand elasticity is less than two in absolute value and as a strategic substitute when greater than two in absolute6value. That is,
∂
∂y

µ

∂Πh
∂x

¶

R 0 according to 2 T |εx | .

The home firm’s price elasticity in demand defined in (11) is the reciprocalof the market share of the home firm denoted by Sx|εx | =

x+y
1
= .
x
Sx

(14)

If its market share is quite large, the individual demand curve the homefirm faces is almost identical with the market demand curve, so the demand
elasticity is close to unity. The home firm behaves like a monopolist. If
the home firm produces a very small part of the market, its market share isvery small and the individual demand curve the home firm faces is eﬀectively
infinite. The home firm behaves like a competitive firm. Thus Proposition 2
can be put in another way: good x of the home firm is a strategic complement(respectively substitute) to good y of the foreign firm when Sx is greater(respectively less) than 0.5. The market share reflects the extent to which afirm dominates the market. The next question which we naturally raise isunder what economic circumstances the home firm can dominate the market?

2.3

Cournot Equilibrium

The intersection of two reaction curves determines the Cournot equilibrium.4
By solving simultaneously two equations of (3) for x and y, we have theequilibrium outputs of the home and foreign firms:
xc =

cy
,
(cx + cy )2

cx
,
y =
(cx + cy )2

(15)

c

and the equilibrium pricepc = cx + cy ,4(16)

We should introduce some expectation formations, construct the dynamic process of
Cournot output adjustment and then define the Cournot equilibrium as a fixed point of
the dynamic process. Instead, we assume the Cournot equilibrium is stable and focus on
the static analysis in this paper. Since Puu’s complex duopoly model generates interesting
dynamics involving chaos (See Puu [2000]), we discuss the case of unstable equilibrium ina diﬀerent paper.

7

where superscript c is attached to the variables associated with the Cournot
equilibrium.5 To see what determines the relative magnitude between xc andy c , we divide the first equation by the second in (15) to get
yc
= k,
xc

(17)

where k is the ratio of the marginal production cost of the home firm overthe marginal production cost of the foreign firm,
cx
(18)
k= .
cy
Thus we find that1
k S 1 =⇒ xc R y c and thus Sxc R ,2(19)

where Sxc is the market share of the home firm at the Cournot equilibrium.
From (17), (18) and Proposition 2, we can say that the home firm regards xc
as a strategic complement if k < 1 and a strategic substitute if k > 1.
This is confirmed geometrically in Figure 1(A). The Cournot equilibrium
satisfies (15) and locates on the reaction curve. We see that the x = y locus
passes through the peaked point, (yM , xM ), of the home firm’s reaction curve.
Thus if the xc = k1 y c locus is flatter than the x = y locus, then it crosses thedownward sloping part of the reaction curve and then xc becomes a strategic
substitute to y c because y c > xc at the Cournot equilibrium, as illustrated
by the dashed line in Figure 1(A). On the other hand if the locus is steeper,xc becomes a strategic complement to y c because y c < xc . Thus, the lastresult and (13) imply the following: output of the home firm is a strategic
substitute for 1 < k, i.e., cx > cy , and a strategic complement for 1 > k,i.e., cx < cy . By the same principle, the foreign firm regards y as a strategic
substitute for 1 > k, i.e., cx < cy and a strategic complement for 1 < k, i.e.,cx > cy . We call a firm with the lower marginal cost of production relatively
eﬃcient and a firm with the higher marginal cost relatively ineﬃcient. Then
we can summarize these results concerning sources of strategic substitutes
and complements as follows.
Proposition 3 A relatively eﬃcient firm regards its own output as a strategic complement and a relatively ineﬃcient firm regards its own output as a
strategic substitute.
5

Mound-shaped reaction curves starting at the origin implies that curves cross eachother twice at (0, 0) and (xc , y c ). The former is the trivial equilibrium point and the latteris the non-trivial equilibrium point. Our concern is with the non-trivial point and thus nofurther consideration is given to the trivial point.

8

Figure 2 visualizes the results of Proposition 3. In Figure 2(A) the homefirm is relatively eﬃcient and the foreign firm is relatively ineﬃcient as theratio of marginal costs is less than unity, k = 0.25. It can be seen that thereaction curve of the home firm is upward sloping (i.e., x is a strategic complement) and the reaction curve of the foreign firm is downward sloping (i.e.,
y is a strategic substitute) in the neighborhood of the Cournot Equilibrium.
In Figure 2(B) where the relative eﬃciency is reversed as k = 5, we observe
the opposite results to the one observed in Figure 2(A) that is x is a strategic substitute and y is a strategic complement in the neighborhood of theCournot Equilibrium.
yym

rf HyLyc

r hHxL

y

rhHxL

C

yc

rf HyL

C

yM

xm

x
xh

xc

xc

(A) cx = 1, cy = 4xxM(B) cx = 1, cy = 0.2.

Figure 2 Reaction Curves and Cournot Equilibrium
Next, we derive the firms’ equilibrium profits calling them the Cournot
profits, which can be calculated as well by substituting xc and y c of (15) into
Πh and Πf of (2),
¶2
µ
cy
c
Πh =
,
cx + cy
(20)
¶2
µ
c
x
Πcf =
.
cx + cy

From (20), we find that the ratio of the Cournot profits is the square of theCournot output ratio,
µ ¶2
1
Πch
=
.
(21)
c
Πf
k
Therefore, from (19) we havek S 1 =⇒ Πch T Πcf
9(22)

(19) and (22)imply the following.
Proposition 4 In a duopoly market with hyperbolic inverse demand andconstant marginal costs, a relatively eﬃcient firm produces larger output andmakes a larger profit than a relatively ineﬃcient firm at the Cournot equilibrium.
Proposition 4 can be alternatively put. From (10), (14) and the profit
maximization condition (i.e., MR = MC), we derive the well-known resultthat if the price elasticity of the market is constant, the price-cost margin isgreater for the firm with larger market sharep − ciSi =for i = x, y.
pThis implies that at a given price, a firm with lower marginal cost (i.e., arelatively eﬃcient firm) produces more than the one with higher marginal
cost, and if marginal costs are equal among firms, then firms produce thesame level of output and thus have the same market share.

3

Strategic Trade Policy

The Cournot equilibrium is the best choice for both firms in the absenceof government intervention as neither firm has an incentive to change its
output decision. As shown in Brander and Spencer [1985] and Eaton and
Grossman [1986], the domestic government can raise domestic welfare byshifting oligopoly profits from the foreign to the domestic firm. These results
are strictly related to whether the goods are strategic substitutes or strategic
complements. In the former, goods are assumed to be strategic substitutes
and the optimal trade policy is to give a home firm an export subsidy. In thelatter, strategic complements are assumed and so the optimal trade policy is
an export tax. The natural question which we should raise is whether active
government intervention is still beneficial in the nonlinear duopoly setting.
In this section, we analyze how intervention by the home governmentaﬀects the market. A familiar two-stage game is set up: in the first stage,
the home government decides the optimal subsidy level, and in the second
stage, firms compete in quantities as in Section 2 taking the subsidy levelas given. As the home government behaves as if a Stackelberg-leader andthe firms Stackelberg-followers we call this game a Stackelberg competition.
The model is solved by backward induction. That is, we first solve for the
Cournot equilibrium output of the two firms in the second stage, and then wesolve for the optimal level of subsidy in the first stage. The solution concept
is subgame perfect Nash equilibrium (henceforth, SPNE).
10

3.1

Stackelberg-leader government and subsidy

First, we look into the second stage and we solve the Cournot equilibrium
output when the home government intervenes in the market. Given the levelof subsidy introduced by the home government, firms engage in a similar
Cournot-Nash duopolistic competition as in the previous section, except thatthe profit of the home firm is aﬀected by the subsidy. Suppose that the homegovernment puts a subsidy s per unit of output produced by the home firm
which reduces the marginal production cost of the home firm. Though theprofit maximization problem of the foreign firm defined in (2) is not aﬀectedby the home government intervention, the profit maximization problem of
the home firm becomes
Max p(x, y)x − (cx − s)x,
where cx − s > 0 is assumed. From the first order conditions of each firms’
maximization problem, the home and the foreign firms’ reaction functions
are
r
y
− y,
x=
cx − s
(23)
r
x
y=
− x.
cy

We can see that the characteristics of the home firm’s mound-shaped reaction
curve are not aﬀected by s, which is the same as that obtained in the previous
section. It starts at the origin. However, a simple calculation yields that its
1
1
from
obtained in the caseintercept on y-axes shifts upward bycx − scx
swithout subsidy. Moreover, the shifted critical values (xsM , yM), at which thereaction curve has the peak, are derived by the substitution of y = xsxsM = yM=

1,4(cx − s)(24)where superscript s means the case with a subsidy. Since the reaction function of the foreign firm is the same as that derived in Section 2, the criticals), are the same asvalues of the peaked point (xsm , yms=xsm = ym

1
4cy

(25)

By solving two equations of (23) simultaneously for x and y, we have the11

Cournot-Nash equilibrium outputs under a subsidy:
xcs =

cy
,
(cx − s + cy )2

cx − s,y cs =(cx − s + cy )2(26)

where superscript cs means the Cournot equilibrium under a subsidy. The
ratio of Cournot outputs under a subsidy isy cs
cx − s=,csxcy
and the price that is determined by the Cournot outputs is similarly derived
aspcs = cx − s + cy .

3.2

Subsidy Eﬀect on the Market

We analyze the impact of a subsidy on the market in the nonlinear setting.
To do this we diﬀerentiate each equation of (26) with respect to s to get
∂xcs
2cy
=
,
∂s
(cx − s + cy )3
cx − s − cy
∂y cs
=
.
∂s
(cx − s + cy )3(27)First we observe that the subsidy has a positive eﬀect on the Cournot outputof the home firm,
∂xcs
(28)
>0.
∂s
That is, if the home government increases (decreases) the subsidy, then thereaction curve of the home firm shifts to the right (left).
Second, we can see that an increase of the export subsidy increases thetotal volume of the exports to the third country regardless of the foreign
firm’s reaction,∂xcs ∂y cs
1
+
=
> 0.
(29)
∂s
∂s
(cx − s + cy )2Third, we see that the eﬀect on the Cournot output of the foreign firm is incs
general ambiguous as the sign of (cx − s) − cy in ∂y∂s can be either positive12

or negative. In particular, depending on the relative magnitude of those twoterms, we have∂y cs
< 0 if ks < 1,
∂s
(30)
∂y cs
> 0 if ks > 1,∂swhere ks is a ratio of firms’ marginal costs after the government intervention
which is defined asks =cx − s.
cy(31)

As the characteristics of the reaction curves of the home firm and the foreign
firm are the same as those obtained in the case without subsidy, Proposition
1 and Proposition 2 are applicable to the case with a subsidy. With the same
reasoning, the foreign firm regards its output as
strategic substitute if ks < 1,
strategic complement if ks > 1.
Therefore it can be said that the subsidy s positively or negatively aﬀects output of the foreign firm according to whether y cs is the strategic complement
or the strategic substitute.
The eﬀect on home output is positive. This result is the same as thatin the linear duopoly setting. However, contrary to the negative eﬀect in
the linear setting, the eﬀect on foreign output can also be positive.6 We
summarize these results.
Proposition 5 The subsidy of the home government has a positive eﬀect on
output of the home firm while it has either a negative eﬀect on the foreign
firm’s output if the foreign output is a strategic substitute or a positive eﬀect
if the foreign output is a strategic complement. Further, regardless of thecharacterization of the related goods of the foreign firm, it has a positiveeﬀect on the total volume of exports to the third country.

6

In the Cournot-Nash duopoly competition with the linear inverse demand function,
the equilibrium output of the home firm increases with subsidy by sacrificing the foreign
firm’s market share.

13

3.3

Subsidy Eﬀect on Profits

We can also analyze the impact of the subsidy on Cournot profits. Inserting
the Cournot equilibrium output (26) into firms’ profit functions yields the
Cournot equilibrium profits under the subsidy:
¶2
µ
cy
cs
Πh =
,
cx − s + cy
(32)
¶2
µ
cx − sΠcs.
f =cx − s + cy
The usual diﬀerentiating procedure produces changes in the equilibrium profits caused by the government strategy,
2c2y
∂Πcs
h
=
> 0,
∂s
(cx − s + cy )3

(33)

∂Πcs
2cy (cx − s)
f
< 0.
=−
∂s
(cx − s + cy )3
The subsidy has a positive eﬀect on the home firm’s profit. However, thehome government subsidy has a negative eﬀect on the foreign firm’s profit.

3.4

Optimal Subsidy

In this subsection we look into the first stage. Given the equilibrium output
in the second stage, the home government now has to decide the optimal levelof subsidy. This is why the home government is called a Stackelberg-leader.
As the home government is assumed to maximize the home country’s welfare,
we define the welfare of the home country as the total sum of the profit of
the home firm less total subsidy:
W = {R(xcs , y cs ) − (cx − s)xcs } − sxcs ,(34)

where R denotes the revenue. Substituting (26) for (34) and diﬀerentiating
it with respect to s yields
¶ cs
µ
dW
∂x
∂R
∂R ∂y cs
=
− cx
+
(35)
ds
∂x
∂s
∂y ∂s
Taking account of the first-order condition for the profit maximization of thehome firm,∂R− (cx − s) = 0,∂x14and the first-order condition for the government’s maximization problem,dW= 0, yields
ds∂R ∂y cs /∂s
,
(36)
s=
∂y ∂xcs /∂s
where
−xcs
∂R
=
,
∂y
(xcs + y cs )2∂y c /∂scx − s − cy.
=∂xc /∂s
2cy
Inserting (26) into (36) and solving it for s gives the optimal level of subsidy,s∗ = cy − cx .

(37)

So we have s∗ S 0 =⇒ k S 1, which is equal to Proposition 1 in Bandyopadhyay (1997). Depending on the firm’s relative competitiveness, the optimal trade policy becomes either an export subsidy, laissez-faire, or an export
tax. Before government intervention, if the home firm is more eﬃcient than
the foreign firm, cy > cx , the home firm regards x as strategic complements
and the foreign firm regards y as strategic substitutes. As a result, the optimal policy is an export subsidy, s∗ > 0. By the same token, if the home firm
is less eﬃcient than the foreign firm, the home firm regards x as strategic
substitutes and the foreign firm regards y as strategic complements so thatan export tax is optimal, s∗ < 0. Only when the two firms have the same
marginal cost, no intervention by the home government is optimal, s∗ = 0.

3.5

Stackelberg Equilibrium

Now that we have the optimal value of the home government’s trade policy,
the SPNE outcomes in the Stackelberg competition can be derived by inserting the optimal level of subsidy s∗ for the equilibrium output in the second
stage. Then, by inserting (37) into (26), (32), and (34), we have the SPNE
output of each firm as well as other SPNE outcomes: the output of each firmiscy
2cx − cyxS = 2 and y S =
,
(38)
4cx
4c2x
associated profits of each firm and the optimal welfare of the home country
are
¶2
¶2
µ
µ
cy
2cx − cycy
SS, Πf =and W S =
,
(39)
Πh =
2cx
2cx
4cx
15

where superscript S attached to the variables implies the Stackelberg equilibrium. Note that when the foreign firm is less competitive at the Stackelberg
equilibrium to such an extent that 2cx − cy ≤ 0 holds, then the foreign
firm does not export and thus the home firm monopolizes the third country
market. To avoid such an interesting case, we assume 2cx − cy > 0 hereafter.
If we substitute the optimal subsidy s∗ into (31), the ratio of the marginal
costs in the case with a subsidy isks∗ =2cx − cy
R 1 ⇔ k R 1.
cy

(40)

provided 2cx − cy > 0. Thus the economic condition to define the strategic
related goods in the case with a subsidy is the same as that in the casewithout subsidy. From (38) and (39), the ratios of Stackelberg outputs andStackelberg profits are given by
ΠSf
yS
∗
=
k
and
=
s
xS
ΠSh

µ S ¶2
y
.
xS

(41)

According to (40), we can have a similar result to the Stackelberg equilibriumin the case with an optimal subsidy with the one at the Cournot equilibrium
obtained in the case with no subsidy summarized in Proposition 4,
if k R 1, then xS S y S and ΠSh S ΠSf .

(42)

For a graphical understanding of the optimal trade policy, we illustrate thereaction curves before and after the government intervention and the isoprofit
curves of the home firm. The Cournot-Nash equilibrium is denoted by C andthe Stackelberg equilibrium by S in Figures 3 and 4 where the pictures in
the right figure are enlargements of those in the left figure. Reaction curves
with no subsidy are depicted with solid lines while the reaction curves withthe optimal subsidy are depicted with bold dotted lines in both figures. Twoisoprofit curves of the home firms are illustrated with normal dotted lines in
the right figure. There, one locus is horizontal at the Cournot equilibrium
and the other touches the reaction curve of the foreign firm at the Stackelberg
equilibrium. It can be observed that the isoprofit curve passing through the
Cournot equilibrium is located above the isoprofit curve passing through the
Stackelberg equilibrium, which implies the profit of the home firm after thegovernment intervention is higher than the profit before. Alternatively put,
the Stackelberg profit of the home firm is higher than the Cournot profit.
In Figure 3, we depict the case in which the home firm is relatively eﬃcientto the foreign firm, i.e., k < 1. According to Proposition 3, x is a strategic
16

complement and y is a strategic substitute.7 Subsidizing the home firm
∂xcs
shifts the reaction curve of the home firm rightward. Since> 0 and∂scs∂y< 0 by (33), its eﬀect is to encourage more production of x and less∂s
production of y. In consequence, the welfare of the home country is increased
at the Stackelberg equilibrium as compared to the Cournot equilibrium as
illustrated in the right figure in which the isoprofit curve is shifted downward.
y

y

yhs

yC

C

yh
S

yB

C

S
xh

x

xC

(A)

xB

x

(B)

Figure 3. Positive Subsidy for (SC,SS) where cx = 2 and cy = 3.
In Figure 4, we depict the case in which the home firm is relatively ineﬃcient as compared to the foreign firm, i.e., k > 1. According to Proposition 3
again, x is a strategic substitute and y a strategic complement. Since the positive subsidy shifts the reaction curve of the home firm rightward so that thehome firm’s isoprofit curve crosses at a higher point than C, which reduces
the home country’s welfare. Thus, it is optimal for the home government to
restrict the production of the ineﬃcient home firm by levying it an export
tax. The export tax shifts the reaction curve of the home firm leftward, sothat the Stackelberg equilibrium S is placed at the lower position than C.
7(SC, SS) in Figure 3 means that the home firm regards its output as a strategic
complement, and the foreign firm’s a strategic substitute. (SS, SC) in Figure 4 below isdefined in the same manner.

17

As shown in (33), we have

∂xcs
∂y cs
> 0 and
> 0.
∂s
∂s

y

y

yh
yhs
C

yC

S

yB

C

xh

x

S

xB

(A)

xC

x

(B)

Figure 4. Export tax for (SS,SC) where cx = 2 and cy = 1.
However, it is true that a production subsidy can generate an increasein exports that possibly reduces welfare of the home country. The general
conclusion from the graphical analysis is that a positive subsidy benefits thehome firm when the foreign firm regards its output as a strategic substituteand is harmful to the home firm when the foregin firm regards its output asa strategic complement.

4

Concluding Remarks

In the international duopoly models, Brander and Spencer [1985] and Eaton
and Grossman [1986] derive opposite policy implications for an export subsidy and an export tax, respectively. This is because in the former, products
are exogenously assumed to be strategic substitutes and, for the latter, strategic complements. In this paper, we introduced a hyperbolic function andexplained that whether products are strategic substitutes or strategic complements depends on demand elasticity. Under a unit elastic demand curve,
we showed that a firm’s strategic behavior is endogenously determined by afirm’s relative competitiveness, which relates to a firm’s mound-shaped reaction curves. We also derive the conditions where the optimal trade policy
will be an export subsidy or an export tax for the home firm.
As mentioned in footnote 4, the stability assumption for our model is critical to our results and we deal with unstable equilibria in a subsequent paper.
Furthermore, a bilateral policy game before the duopolistic competition in
the good market must also be investigated.
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References[1] Bandyopadhyay, S., ”Demand elasticities, asymmetry and strategic
trade policy,” Journal of International Economics, 42, 167-77, 1997.
[2] Brander, J., and B. Spencer, ”Tariﬀ and the Extraction of Foreign
Monopoly Rents under Potential Entry,” Canadian Journal of Economics, 14, 371-389, 1981.
[3] Brander, J., and B. Spencer, "Trade Warfare; Tariﬀs and Cartels," Journal of International Economics, 16, 227-242, 1984a.
[4] Brander, J., and B. Spencer, "Tariﬀ Protection and Imperfect Competition," in Monopolistic Competition and International Trade, edited by
Henryk Kierzkowske, Osford:Clarendon Press, 194-206, 1984b.
[5] Brander, J., and B. Spencer, "Export Subsides and International Market
Share Rivalry," Journal of International Economics, 18, 83-100, 1985.
[6] Bulow, J., Geanakoplos, J., and Klemperer, P., “ Multimarket
Oligopoly: Strategic Substitutes and Complements,” Journal of Political Economy, Vol.93, pp.488-511, 1985.
[7] Eaton, J. and G. Grossman, ”Optimal Trade and Industry Policy under
Oligopoly,” Quarterly Journal of Economics, Vol.101, pp.383-406, 1986.
[8] Jones, R.W., ”Trade taxes and subsidies with imperfect competition,”
Economic Letters, 23, pp.375-379, 1987.
[9] Matsumoto, A. and N. Serizawa, "Nonlinear International Trade,"mimeo, 2003.
[10] Puu, T., Attractors, Bifurcations, & Chaos, Springer-Verlag, Berlin, Heidelberg, 2000.

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