A Little Help from My Friend: International
Subsidy Games with Hyperbolic Demands∗
Akio Matsumoto†Chuo University

Ferenc Szidarovszky‡
University of Arizona

Abstract
In this paper, we assume a hyperbolic price function and constructa three-country model with two active governments and two firms. The
purpose of this study is to consider dynamic behavior of the sequential
subsidy game in which the governments determine their optimal trade
policies and, accordingly, the firms choose their optimal outputs. We firstshow the existence of an optimal trade policy under realistic conditions.
Our main results are summarized as follows: (1) when the production costs
are identical, then a trade policy and the corresponding optimal output
are stable if the demand is elastic while multistability (i.e., coexistence ofmultiple attracors) and complex dynamics occur if the demand is inelastic;(2) when the production costs are diﬀerent, then a stable trade policy
induces chaotic output fluctuations regardless of demand elasticity; (3)policy dynamics can be chaotic if demand is elastic while multistability
still occurs if the demand is inelastic.

Keywords: Three-country model, Two-stage game, Policy dynamics,
Chaos, Multistability, Hyperbolic price function

JEL Classification: F13∗ This paper was prepared when the first author visited the Department of Systems andIndustrial Engineering of the University of Arizona. He appreciated its hospitality over his
stay. The authors are grateful to Junichi Minagawa for his supporting computational study.
The usual disclaimer applies.
† Department of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo, 1920393, Japan. akiom@tamacc.chuo-u.ac.jp‡ Department of Systems and Industrial Engineering, University of Arizona, Tucson, AZ,
85721-0020, USA. szidar@sie.arizona.edu

11Introduction

Market is imperfectly competitive since the number of firms is small, the goods
are diﬀerentiated or there is some kind of economies of scale. In real world, we
see many imperfectly competitive industries in which the firms compete fiercely
both domestically and internationally. In such a imperfectly competitive international market, governments may have incentives to introduce trade policies
like tariﬀ, export subsidy and tax. If governments can aﬀect, or more precisely,
strategically alter market structure by introducing polices, then it is very important to examine how behavior of the firms would be aﬀected or what the
impact it would have on the market. To address these issues, a lot of eﬀorts
have been devoted since the 1980s. Brander and Spencer (1985) demonstratethat an increase in a domestic export subsidy raises the domestic profit when
the firms compete in a Cournot way. Eaton and Grossman (1986) show that an
export tax can be optimal when the firms compete in a Bertrand way. It is now
well-known that the source of this sharpe diﬀerence comes from the diﬀerence inthe assumption on the firms’ strategic behavior (that is, the goods are strategic
substitutes or strategic complements, the definitions of which are due to Bulow
et al. (1985)). It is also well-known that this behavioral diﬀerence relates tothe curvature and the elasticity of the demand function. Recently, constructing
a simplified three-country model with two firms and two governments, Bandyopadhyay (1997) shows the following clear-cut results on the trade policy when
demand is hyperbolic:(1) When the production costs of the firms are identical, then a subsidy is
optimal if demand is elastic while a tax is optimal if demand is inelastic.
(2) When the production costs are diﬀerent, then the lower-cost firm enjoys
the higher subsidy and such a trade policy is locally stable in policy space
if demand is elastic while the higher-cost firm enjoys the higher subsidy
and such a policy is locally unstable if demand is inelastic.
We move one step forward to consider local as well as global dynamics oftrade policy and corresponding outputs in the three-country model. This study
complements the results of Bandyopadhyay from a dynamic point of view. It
is also an extension of the work of Matsumoto and Serizawa (2007) who focus
mainly on the comparative static analysis of a similar three-country model. The
price function is assumed to be hyperbolic so that the dynamic model of outputs
to be considered in this paper resembles nonlinear dynamic duopoly models,
which has been extensively studies during the last twenty years. Comprehensive
summary of the earlier works has been presented in Puu and Sushko (2002) and
Puu (2003). More recent developments on this filed are given in Bischi et al.
(2009). The third aims of this study is to apply the theoretical results obtained
there to the dynamic analysis in the framework of international economics.
This paper is organized as follows. Section 2 presents a variant of the threecountry model in which both governments take active roles. Section 3 considers
policy dynamics and the corresponding output dynamics when the demand is2

elastic. Section 4 examines the same dynamic issue when the demand is inelastic.
Section 5 gives concluding remarks.

2

The Model

The model presented below is a variant of the three-country model. Two countries are called Home and Foreign and governed by the Home government
and the Foreign government (henceforth referred to as H-government and F government). There is one firm in each country, the firm in the Home country
is named firm 1 and the one in the Foreign country firm 2. They produce indifferent outputs, x and y, with constant marginal costs, c1 and c2 , respectively,
and export all of the outputs they produce to a third country. Competition in
the third country is modeled through a two-stage game. At the first stage, thegovernments hosting their firms choose subsidy rates, si for i = 1, 2, so as tomaximize their welfare, taking the optimal behavior of the firms as given. At
the second stage, the firms employ the quantity competition in a Cournot way
and choose outputs so as to maximize their profits, taking their governments’
trade policies as given. Optimal subsides and optimal outputs are backwardly
determined.
This section is divided into four parts. We first solve the profit maximization
problems of the firms in Section 2.1 and then in Section 2.2 we consider stabilityof the output equilibrium, taking the trade policies of the governments as given.
We solve the welfare maximization problems of the governments in Section 2.3
and finally determine the optimal trade policy in Section 2.4.

2.1

Profit Maximization

Let the inverse demand function be hyperbolic,
P =

1
,
Qλ

where Q is the total output, Q = x + y, λ is the reciprocal of demand elasticity
and λ > 0. At the second stage in which governments’ subsidies are given, firm
1 and firm 2 choose outputs to maximize their profits
π1 = (P − (c1 − s1 ))x,and
π 2 = (P − (c2 − s2 ))y.

The first-order conditions of the profit maximization are given by(1 − λ)x + y∂π 1− cx = 0,=∂x(x + y)λ+1
and(1 − λ)y + x∂π 2− cy = 0,=∂y(x + y)λ+1
3

where cx = c1 − s1 and cy = c2 − s2 for notational simplicity.1 We call the production cost including the subsidy an actual cost. Although we will show laterthat the actual costs with the optimal subsidies are non-negative, we suppose
for a time being that cx > 0 and cy > 0. From the first-order conditions, theimplicit forms of firms’ best responses are derived as(1 − λ)x + y = cx (x + y)1+λ ,(1)(1 − λ)y + x = cy (x + y)1+λ .

(2)

and
Dividing (1) by (2) yieldsy=(1 + k)λcx
k − (1 − λ)x, x + y =x where k = .
1 − k(1 − λ)1 − k(1 − λ)cyThe first two equations are substituted into (1) and (2), and we solve the resultant equations for the corresponding outputs to obtain1xC =(2 − λ) λ (cy − (1 − λ)cx )yC =(2 − λ) λ (cx − (1 − λ)cy )λ+1λ(cx + cy ) λ

and

,

(3)

,

(4)

1

λ+1

λ(cx + cy ) λwhere superscript C is attached to variables associated with the Cournot point.
In order to assure the nonnegativity of Cournot outputs, we make the following
assumptions.
Assumption 1. (1) 0 < λ < 2 and (2) 1 − λ < k <1when λ < 1.
1−λFor the convenience of latter considerations, we define by Ω the feasible setof demand elasticity and the actual cost ratio that satisfy Assumption 1,Ω = {(λ, k) | 0 < λ < 2, and 1 − λ < k <1for λ < 1}.
1−λ

The Cournot outputs in (3) and (4) are substituted into the profit functions to
obtain the Cournot profits:
1−λ

πC
1 =

(2 − λ) λ (cy − (1 − λ)cx )2,
1+λλ(cx + cy ) λand(2 − λ)
πC
2 =
λ

1−λ
λ

(cx − (1 − λ)cy )2
1+λ(cx + cy ) λ

.

(5)

(6)

1 It can be checked that the second-order conditions are satisfied for any x and y that solve
the first-order conditions.

4For later analysis, we point out that
⎧
"
#
∙ C¸
µ
¶
⎪
∂ ∂π 1∂x⎪⎪sign= sign
,
⎪
⎪
⎪
∂y
∂x
∂s2
⎨
|(x,y)=(xC ,y C )
#
"
⎪
∙ C¸
µ
¶
⎪
⎪
∂y
∂π
∂
⎪
2
⎪
=
sign
,
sign
⎪
⎩
∂x ∂y |(x,y)=(xC ,yC )
∂s1

(7)

where the left hand sides are the signs of the cross derivatives of the marginal
profit functions evaluated at the Cournot point and the right hand sides are thecross eﬀects of the Cournot outputs caused by a change in the subsidy of therival government, for instance,
∂
∂y

µ

∂π 1
∂x

¶

=

cy
(xC + y C )2+λµ2−λ(cx + cy )1+2λ|(x,y)=(xC ,y C )

and,
∂xC
cy
= 2
∂s2
λ

µ

2−λ
(cx + cy )1+λ

¶ λ1

¶ λ1

¢
¡
1 − (1 + λ − λ2 )k

¢¡1 − (1 + λ − λ2 )k .

The terms in the square brackets in the left hand side of (7) refers to thedefinitions of a strategic substitute and a strategic complement. The output offirm 1 is strategic substitutes or strategic complements according to whether
1 − (1 + λ − λ2 )k is negative or positive. In the similar way, it can be shownthat the output of firm 2 is a strategic substitute and a strategic complement
according whehter k − (1 + λ − λ2 ) is negative or positive. It is not diﬃcult toshow that an increase of the subsidy of one government increases the output of
its firm. Hence the equlity in (7) implies that if the output of firm 1 is a strategic
substitute to the output of firm 2, then an increase of the subsidy given by F government decreases the output of firm 1 via increasing the output of firm 2.
In the same way, if the output of firm 1 is a strategic complement to the output
of firm 2, then an increase of the subsidy given by the F -government increases
the output of firm 1 through increasing the output of firm 2. Figure 1 shows thedevisions of the feasible set Ω by the stragegic characteristics of the outputs.2
The set Ω is shaded in gray and is divided into the four parts by two curves,
k = 1 + λ − λ2 and k(1 + λ − λ2 ) = 1. There, ”S” and ”C” stand for strategic
substitute and strategic complement. Thus the region denoted by (S, C) meansthat firm 1 treats its output as a strategic substitute whereas firm 2 considers its
output as a strategic complement for (λ, k) belonging to the region, and another2 We refer to λ∗ and the points denoted by A and a later.

5

combinations are defined in the same way.

Figure 1. Division of feasible (λ, k) region Ω

2.2

Output Dynamics

We turn our attention to an output adjustment process. We first derive local stability conditions and then consider the nonnegativity conditions for the output
trajectories. From (1) and (2), the adjustment process with native expectations
can be written as the implicit equations,⎧⎨ (1 − λ)x(t + 1) + y(t) = cx (x(t + 1) + y(t))1+λ ,(8)⎩x(t) + (1 − λ)y(t + 1) = cy (x(t) + y(t + 1))1+λ .

The first equation will be denoted by γ 1 (x(t + 1), y(t)) = 0 and the second
equation by γ 2 (x(t), y(t + 1)) = 0 as it is impossible to solve explicitly (8) forx(t + 1) and y(t + 1) unless λ = 1.3
The fixed point of this process has been already obtained in (3) and (4). To
find local stability conditions, we derive the Jacobi matrix by linearizing equations of (8) in the neighborhood of the Cournot point and locate the eigenvalues.
Notice that the Jacobi matrix has the special form,
⎛

⎜
J =⎜
⎝

0
(λ2 − λ − 1) + k(1 + 2λ − λ2 ) − (1 − λ)k⎞(λ2 − λ − 1)k + 1(1 + 2λ − λ2 )k − (1 − λ) ⎟⎟.
⎠0

3 A two-stage game with unit-elastic demand (i.e., λ = 1) is considered in Matsumoto andSzidarovszky (2009).

6

Since trace of the Jacobian matrix is zero, stability is confirmed if the absolutevalue of the product of the eigenvalues is less than unity:¯¯¯¯(λ2 − λ − 1)k + 1(λ2 − λ − 1) + k¯ < 1. (9)|Ψ(λ, k)| = ¯¯−22(1 + 2λ − λ )k − (1 − λ) (1 + 2λ − λ ) − (1 − λ)k ¯

Solving Ψ(λ, k) = −1 yields k = −1, λ = 0 or λ = 2. They contradict toAssumption 1 and to the fact that the actual cost ratio is positive. We omit thiscase for further considerations. Solving Ψ(λ, k) = 1 for k yields two solutions
for which loss of stability occurs:
´
³
p
⎧
⎪
2 + λ 2 + λ − 3λ2 + λ3 − (λ + 1)(λ − 2) λ2 − 4λ + 5⎪⎪⎪
ψ (λ) =,⎪⎪⎨ 1
2 + λ(2 − 4λ + λ2 )
(10)
´
³
p
⎪
⎪
2
3
2
⎪
2 + λ 2 + λ − 3λ + λ + (λ + 1)(λ − 2) λ − 4λ + 5⎪⎪⎪⎩ ψ 2 (λ) =.
2 + λ(2 − 4λ + λ2 )For all k above the k = ψ 1 (λ) curve and below the k = ψ 2 (λ) curve, the absolutevalue |Ψ(λ, k)| is greater than one. so the Cournot point is locally unstable. In
the same way, for all k between these curves, the absolute value |Ψ(λ, k)| is lessthan unity implying the local asymptotic stability of the Cournot point.
Next we examine the nonnegativity of trajectories generated by (8). From
the first equation of (8), it can be found that x(t + 1) = 0 for y(t) = ymax where
ymax =

µ

1
cx

¶ λ1

.

It is also found that the output x(t + 1) arrives at its maximum xmax for y(t) =ym whereµ¶ λ1
1xmax =and ym = λxmax
cx (1 + λ)1+λ1+λThe relation xmax + (1 − λ)ymax = cy (xmax + ymax )describes that firm 2chooses to produce ymax when it expects that the competition will produce xmax .
Arranging this equation yields
¶ λ1
1
1−λ+
(1 + λ)1+λk = φ1 (λ) = Ã.
µ¶ λ1 !1+λ1
1+(1 + λ)1+λµ(11)

For k < φ1 (λ), a trajectory of output x can be negative. So k ≥ φ1 (λ) is thenonnegativity condition for the trajectories when the actual cost ratio is less7

than unity. In the same way we can obtain the nonnegativity condition for the
trajectories when the actual cost ratio is greater than unity as k ≤ φ2 (λ) where

φ2 (λ) =¶ λ1 !1+λ
11+µ¶(1 + λ)1+λ
1
=
.
µ
¶ λ1
φ1 (λ)11−λ+(1 + λ)1+λ

Ã

µ

(12)

We denote the unstable and nonnegativity region byΩU = {(λ, k) ∈ Ω | φ1 (λ) < k < ψ 1 (λ) or ψ 2 (λ) < k < φ2 (λ)}.
Whenever a pair of (λ, k) falls inside region ΩU , the Cournot point becomes
locally unstable. Although the local instability means global instability in a
linear dynamic model, this is not a case in the nonlinear case. We examine
what kind of dynamics the output adjustment process (8) can generate in thecase of local instability. However, equation (8) is given in implicit forms, so it
is diﬃcult to examine dynamics analytically. At the expense of generality, we
perform numerical simulations after determining the optimal trade policy of thegovernments.

2.3

Welfare Maximization

At the first stage of the two-stage game, the governments determine the optimal
subsidy levels to maximize the national welfareC
W1 (s1 , s2 ) = π C
1 (s1 , s2 ) − s1 x (s1 , s2 ),(13)C
W2 (s2 , s1 ) = π C
2 (s2 , s1 ) − s2 y (s2 , s1 ),

(14)

and
where for notational simplicity, H- and F -governments will be also indexed by"1" and "2". Our first interest is on the condition under which the government
decides to give a subsidy or to charge a tax. Assuming an interior optimum andsolving the first-order conditions of the welfare maximization for si yielddP ∂y C /∂s1
∂y C
s1 = xC
R 0 according toQ 0,CdQ ∂x /∂s∂s1
| {z } | {z 1}
(−)

and

(+)

dP ∂xC /∂s2∂xC
s2 = y CR0accordingtoQ0
dQ ∂y C /∂s∂s2| {z } | {z 2}
(−)

(+)

8

where P is the price function and Q = x + y the outputs of the industry.4
Since the sign of the cross derivative depends on the strategic characteristic of
the output as shown in (7), the optimal trade policy of the governments are
summarized as follows.
Lemma 1 A government pays a subsidy to its firm if the rival firm considers
its own output as strategic substitute and levies an export tax to its firm
if the rival firm considers its own output as strategic complement.
Our second interest is on subsidy diﬀerential. Neary (1994) shows that thesubsidy diﬀerential can be expressed asξs1 − s2 = − (c1 − c2 )ψwhere, in his notation5 , ξ and ψ are given byξ = 2 + R + αα∗ R2andψ = 2 + R.
Using cx and cy , we can express the subsidy diﬀerential in terms of the actual
cost diﬀerential,ξs1 − s2 = −(cx − cy )ψ+ξwhere ψ + ξ = 2(2 + R) + αα∗ R2 . Neary (1994) has already shown that ξ > 0
and ψ + ξ > 0 due to the second-order conditions of the welfare maximization
problems. Thus we can summarize the result on the subsidy diﬀerential asfollows.
Lemma 2 The government gives a subsidy in such a way that the firm with thelower actual cost receives the higher subsidy than the firm with the higher
actual cost.
Notice that Lemma 1 is concerned with the sign of subsidy and Lemma 2 isconcerned with the subsidy diﬀerential. Based on these lemmas, we can arriveat the following optimal subsidy policy.
4 For instance, the first-order condition for the H-governemnt is
C
C
∂πC
∂π C
∂π C
∂xC
1 ∂x
1 ∂y
1
+
+
− x1 − s1=0∂x ∂s1∂y ∂s1
∂s1
∂s1

where

∂πC
∂πC
∂π C
∂P ∂Q ∂Q
1
1
1
= 0,
= xC
,
= 1 and= xC .
∂x∂y∂Q ∂y ∂y∂s1

5 R is a measure of the concavity of demand curve defined by Q P 00 , α and α∗ are the
P
yC
xC
∗
market share of firm 1 and firm 2 at the Cournot point, respectively, α = Q
C and α = QC
where QC = xC + yC .

9

Theorem 1 The interior optimal subsidies, se1 and se2 , depend on the actual
cost ratio and the strategic characteristics of the outputs in the following way:
(1) If both outputs are strategic substitutes, then the governments give subsidies
to their firms such thatse1 ≥ se2 > 0 if cy ≥ cx and se2 > se1 > 0 if cx > cy .
(2) If both outputs are strategic complements, then the governments charge export taxes to their firms such that0 > se1 ≥ se2 if cy ≥ cx and 0 > se2 > se1 if cx > cy .
(3) In the mixed case when one firm treats its output as a strategic substituteand the other firm treats its output as a strategic complement, then thefirm with the higher actual cost receives an export tax while the firm withthe lower cost receives an export subsidy;
se1 > 0 > se2 if cy > cx and se2 > 0 > se1 if cx > cy .
When c1 = c2 , as seen in Figure 1, the outputs are strategic substitutes if
demand is elastic and are strategic complements if demand is inelastic. Thus
parts (1) and (2) of Theorem 1 can be restated as follows6 :Corollary 1 When the firms are symmetric, an export subsidy is optimal if
demand is elastic and an export tax is optimal if demand is inelastic
In order to get a complete description of the dynamics in the international
subsidy game, we have to specify the welfare functions, derive the explicit formsof the best reply functions of the governments and consider their characteristics
in the policy space. Substituting xC , y C , QC = xC + y C and P C = (QC )−λ into(13) and (14) yields the explicit forms of H-government’s welfare function,
1(2 − λ) λ −1
(cx + cy − c1 (2 − λ)) (cy − (1 − λ)cx ) ,W1 (s1 , s2 ) =1
λ(cx + cy ) λ +1

(15)

and F -government’s welfare function,
1W2 (s2 , s1 ) =(2 − λ) λ −1
(cx + cy − c2 (2 − λ)) (cx − (1 − λ)cy ) .
1
λ(cx + cy ) λ +1

(16)

Diﬀerentiating W1 (s1 , s2 ) and W2 (s2 , s1 ) with respect to s1 and s2 , respectively,
yields the first-order conditions,∂W1= βf (s1 , s2 ) = 0 =⇒ f (s1 , s2 ) = 0,
∂s1
6 These are the same as Case 2a and Case 3a of Bandyopadhyay (1997).

10

(17)

and

∂W2
= βg(s2 , s1 ) = 0 =⇒ g(s2 , s1 ) = 0,∂s2where β is a positive constant,

(18)

1

β=

(2 − λ) λ −1
> 0,
12
λ (cx + cy ) λ +2

and f (s1 , s2 ) and g(s2 , s1 ) are defined, respectively, byandf (s1 , s2 ) = (cx + cy )(cy − s1 − (1 − λ)c1 )(1 − λ)λ¡¢+ cy − (1 − λ)cx )(cy − s1 − (1 + λ − λ2 )c1 ,g(s2 , s1 ) = (cx + cy )(cx − s2 − (1 − λ)c2 )(1 − λ)λ¡¢+ cx − (1 − λ)cy )(cx − s2 − (1 + λ − λ2 )c2 .

The second-order conditions areanddf (s1 , s2 )df (s1 , s2 )∂ 2 W1
=β
<0⇔
< 0,
2
∂s1
ds1
ds1

(19)

dg(s2 , s1 )dg(s2 , s1 )∂ 2 W2
=β
<0⇔
< 0,
∂s22
ds2
ds2

(20)

where the derivatives are

and

n
h
io
df (s1 , s2 )= − 2s1 (1 − λ)2 + λ (3 − 2λ)cy +(1 − λ)c1 ,ds1©
ª
dg(s2 , s1 )= − 2s2 (1 − λ)2 + λ [(3 − 2λ)cx +(1 − λ)c2 ] .
ds2

A best reply function of the H-government is derived first. Solving f (s1 , s2 ) =0 with respect to s1 yields a pair of roots,
o
n
p
1
+
(3
−
2λ)c
]
±
(2
−
λ)
D
(s
)
,
(21)
s1S,L = −
λ
[(1
−
λ)c
1
y
1 2
2(1 − λ)2where s1L is the larger root, s1S is the smaller root and D1 (s2 ) denotes thedicsriminant defined by
D1 (s2 ) = (cy − (1 − λ)c1 )2 + 4(1 − λ)λ2 c1 cy .

It is clear that D1 (s2 ) > 0 if λ < 1. In the case of λ > 1, it is possible todetermine two threshold values of s2 that makes D1 (s2 ) = 0,
n
o
p
(±)
s2 = c2 + (λ − 1) (1 − 2λ2 ) ± 2λ λ2 − 1) c1 .
11

(+)

(−)

It then follows that D1 (s2 ) > 0 for s2 > s2 or s2 < s2 if λ > 1.
Next we examine which root of (21) satisfies the second-order condition.
Supposing D1 (s2 ) > 0, multiplying both sides of (21) by 2(1 − λ)2 and thenmoving the first two terms of the right hand side to the left, we obtain thealternative expressions,p2(1 − λ)2 s1L + λ((1 − λ)c1 + (3 − 2λ)cy ) = (2 − λ) D1 (s2 ),andp2(1 − λ)2 s1S + λ((1 − λ)c1 + (3 − 2λ)cy ) = −(2 − λ) D1 (s2 ).

The left hand sides of both equations are equal to −df (s1L , s2 )/ds1 and −df (s1S , s2 )/ds1 ,which should be positive if the root satisfies the second-order condition. The
right hand side of the first equation is positive and the one of the second equation is negative under the assumptions of λ < 2 and D1 (s2 ) > 0. Hence, given
s2 , s1L is the root that satisfies the first-order and the second-order conditions
and therefore it is the best reply function of the H-government:p
λ [(1 − λ)c1 + (3 − 2λ)cy ] − (2 − λ) D1 (s2 ).
(22)
r1 (s2 ) = −2(1 − λ)2In the same way, we can show that the following solution of g(s2 , s1 ) = 0 withrespect to s2 is the best reply function of the F -government:p
λ [(1 − λ)c2 + (3 − 2λ)cx ] − (2 − λ) D2 (s1 )(23)
r2 (s1 ) = −2(1 − λ)2where D2 (s1 ) is the discriminant defined by
2

D2 (s1 ) = (cx − (1 − λ)c2 ) + 4(1 − λ)λ2 c2 cx .
Similary to the previous case, D2 (s1 ) > 0 if λ < 1. In the case of λ > 1, thecritical values of s1 making D2 (s1 ) = 0 are obtained as
n
o
p
(±)
s1 = c1 + (λ − 1) (1 − 2λ2 ) ± 2λ λ2 − 1) c2
(+)

(−)

where D2 (s1 ) > 0 for s1 > s1 or s1 < s1 if λ > 1.
We will next impose the following two external constraints on the subsidy
levels, si , since the governments must behave with control. The first constraint
is the lower bounds sL
i of the subsidy, which is the upper bound of the export
tax and is negative. Its specific value will be determined later. The second
constraint requires an upper bound sU
i of the subsidy, which may be due tothe limited amount of governments’ budgets, and is assumed to be equal to the
production cost. Intuitively speaking, in determining their export subsidies, thegovernments will not give more than the production costs of their firms.
U
L
U
Assumption 2 sLi ≤ si ≤ si for i = 1, 2 where si < 0 and si = ci > 0.

12

U
If demand is elastic, then r1 (s2 ) restricted to the interval [sL
2 , s2 ] and r2 (s2 )L U
restricted to the interval [s1 , s1 ] are the constrained best reply functions. They
are continuous on these intervals. If demand is inelastic, then the best reply
functions restricted these domains are modified to be piecewise continuous as it
will be shown later.
Let us examine in more detail the shapes of the best reply functions in the(+)(−)
case of λ > 1. r1 (s2 ) is defined for only s2 ≥ s2 and s2 ≤ s2 so it is not(−)(+)
defined for s2 < s2 < s2 . In Figure 2, the downward-sloping dotted-solidbold curve is shown in the lower-left and the slightly upward-sloping bold curvein the upper-right part of the figure. They are two pieces of the s1 = r1 (s2 )curve. The upward-sloping solid line is the df1 (s1 , s2 )/ds = 0 locus. Let sM
1 be
(+)
(+)
the solution of df1 (s1 , s2 )/ds = 0. Then s2 < c2 and c1 < sM
becausefor1
1 < λ < 2,p(+)
s2 = c2 + (λ − 1){(1 − 2λ2 ) + 2λ λ2 − 1}c1 < c2
{z
}
|
(−)

and

sM
1 =λ
|

(

2

r

2λ − λ − 2 + λ(3 − 2λ)
{z

≥1
(+)

)
λ+1
c1 > c1 .
λ−1
}
(+)

The direct substitution of s2 into r1 (s2 ) shows that r1 (s2 ) = sM
1 whileUUsU
1 = c1 and s2 = c2 by definition. It then follows that r1 (s2 ) > s1 for(+)
U
s2 ≤ s2 ≤ s2 . This imply that the best reply function violates Assumption 2(+)
and thus is not defined for s2 ≤ s2 ≤ sU
2 either. To remedy this unfavorable
U
property of r1 (s2 ), we extend it to the interval [sL
2 , s2 ] in the following way.
On the upward-sloping solid curve of Figure 2, 0 = df (s1 , s2 )/ds1 holds and thesecond order condition is satisfied under this curve. Let sm
1 be the solution of(−)
L`
0 = df (s1 , s2 )/ds1 .7 Under the assumption that sm≤s,1
1 s2 is defined so that(−)`
r1 (s`2 ) = sL
1 and s2 ≤ s2 . Then define the piecewise best reply function of the
H-government by⎧`⎨ r1 (s2 ) for sL
2 ≤ s2 ≤ s2 ,R1 (s2 ) =
(24)
⎩ L
.
s1 for s`2 < s2 ≤ sU
2`
In Figure 2, the bold curve kinked at (sL
1 , s2 ) is the best reply curve definedby (24). In the same way, the piecewise continuous best reply function of theL
F -government is also defined under the assumption that sm
2 ≤ s2 :⎧`⎨ r2 (s2 ) for sL
1 ≤ s1 ≤ s1 ,R2 (s1 ) =(25)⎩ L
s2 for s`1 < s1 ≤ sU
1
7 sm = λ
1

u


λ+1
(−)
2λ2 − λ − 2 − λ(3 − 2λ)c1 . It also holds that rH (s2 ) = sm
1 .
λ−1

13

(−)

where s1 is the smaller root that solves D2 (s1 ) = 0, sm
2 is the solution of
(−)
`
`
L
equation df (s2 , s1 )/ds2 = 0 and s1 satisfies r2 (s1 ) = s2 .

Piecewise continuous best reply function of the H-government

2.4

Determination of the Optimal Subsidy

In this section, we look for an explicit solution of the trade policy that is determined by an intersection of the best response curves of the two governments.
The nonlinearity of the best reply functions may lead to multiple optimal points.
Since the specific forms of the best reply functions depend on the cost structureand demand elasticity, we first consider the special case with symmetric firms(i.e., c1 = c2 ) and then proceed to the general case.
Substitute c1 = c2 = c into (22) and (23) and suppose that λ < 1. In Figure
3 in which we take λ = 0.8 and c = 1,8 the reaction curve of the H-government
takes a U -shaped profile with respect to the s2 axis and so does the reaction
curve of the F -government with respect to the s1 axis.9 The area surrounded byU
L U
the dotted rectangle is the feasible region defined by [sL
1 , s1 ]×[s2 , s2 ]. Since thetwo governments are symmetric with respect to the diagonal of the (s1 , s2 ) space
under the assumption of identical production costs, their intersection occurs at
a point on the diagonal. Solving r1 (s2 ) = r2 (s1 ) with the condition s1 = s2
yields two positive solutions, which are denoted by E e and E A in Figure 3.
Two solutions imply that the governments have two choices, the lower (interior)subsidies,λ(1 − λ)c
E e = (se1 , se2 ) with se1 = se2 =
> 0,
(26)
2(3 − 2λ)8 Point (0.8, 1) corresponds to point A in Figure 1.
9 To keep notational consistency, we denote the best reply functions in the case of λ < 1
by Ri (si ) for i, j = 1, 2 and i 6= j.

14and higher (cornered) subsidies,
A
A
U
A
U
E A = (sA
1 , s2 ) with s1 = c1 (= s1 > 0) and s2 = c2 (= s2 > 0).

(27)

Figure 3. Determination of the optimal subsidy in the case of λ < 1
If demand is inelastic (i.e., λ > 1), then there are three optimal points
denoted by E e , E a and E b , as shown in Figure 4 in which we take λ = 1.1 andc = 1.10 The solid curve and the dotted curve are the best reply curves of the
H-government and the F -government, respectively. It can be seen that at aninterior point, E e , the governments charge export taxes,
E e = (se1 , se2 ) with se1 = se2 =λ(1 − λ)c < 0.
2(3 − 2λ)(28)

The piecewise-continuous curves intersect at two other points, E a and E b , which
are born because we modified the best reply functions. The two points describe
a mixed trade policy in a sense that one government chooses to give an export
subsidy and the other government to levy an export tax,aL
E a = (sa1 , sa2 ) with sa1 = sL
1 < 0 and s2 = rF (s1 ) > 0

(29)

b
L
E b = (sb1 , sb2 ) with sb1 = rH (sL2 ) > 0 and s2 = s2 < 0.

(30)

and
1 0 Point (1.1, 1) corresponds to point a in Figure 1.

15

Figure 4. Determination of the optimal subsidy in the case of λ > 1
We now determine the appropriate value of sL
i . The best reply functionsLmL
are obtained under the assumptions that sm
1 ≤ s1 and s2 ≤ s2 , which can bespelled out as
Ã
!
r
λ+1
2
(31)
λ 2λ − λ − 2 − (3 − 2λ)λc ≤ sLi
λ−1
The equilibrium subsidies are constrained by the two conditions: first they must
be greater than the lower bound,λ(1 − λ)c ≥ sLi2(3 − 2λ)(32)

second, the interior equilibrium point is located in the region where the secondorder conditions are satisfied. To this end, the following condition is necessary,λ(1 − λ)cλ(3λ − 4)c2 ≥ 2(3 − 2λ) ,2 − 7λ + 4λ(33)

where the value of the left hand side corresponds to the coordinate of an intercept of the two loci, df (s1 , s2 )/ds1 = 0√ and dg(s√1 , s2 )/ds2 = 0. These three
61−11√ 17
constraint curves cross at the point ( 15−8 17 , 16(3−c) ' (1.36, −0.87c) and17)√(33) is always true for λ ≤ 15−8 17 . If we assume the following assumption, thenthese three conditions are satisfied:√λ(1 − λ)15 − 17∗m
L
Assumption 3. si ≤ si ≤and λ ≤ λ =.
2(3 − 2λ)8
We summarize the above dervation in the following theorem:

16

Theorem 2 Suppose that the firms are symmetric and Assumption 3 holds.
The governments have two distinctive policies of an export subsidy if demand iselastic and three distinctive policies (one is an export tax and the other two are
mixed policies) if demand is inelastic.
Now we proceed to the determination of the optimal trade policy when
the firms are asymmetric (i.e., c1 6= c2 ). In principle, the optimal subsidy isdetermined by an intersection of the best reply curves of the governments. By
continuity, the intersection is located in the first quadrant of the (s1 , s2 ) space
if the cost diﬀerence is small and in the second or fourth quadrant if it is large.
However, due to the complicated expressions of (22) and (23), we cannot derive
general explicit solutions so we specify the parameters’ value and numericallyobtain the intersections when the cost diﬀerence is large. We start with the caseof elastic demand. With c1 = 1 and λ = 0.8 we take c2 = 1.25 in Figure 5(A)where se1 ' 0.49 and se2 ' −0.22. Similary we select c2 = 0.75 in Figure 5(B)where se1 ' −0.16 and se2 ' 0.42. The actual cost ratios arek=

1 − 0.49
1 − (−0.16)' 0.34 and k =' 3.45.
1.25 − (−0.22)
1.25 − 0.42

When (λ, k) = (0.8, 0.34) or (λ, k) = (0.8, 3.45), the government adopt themixed trade policy: one government gives an export subsidy to its firm and theother government levies an export tax on its firm. Notice that the lower-costfirm enjoys the higher subsidy.

(A) c1 = 1 and c2 = 1.25

(B) c1 = 1 and c2 = 0.75

Figure 5. Asymmetric firms and elastic demand (λ = 0.8)
We turn to the case of inelastic demand. With c1 = 1 and λ = 1.1, we takec2 = 1.05 in Figure 6(A) where se1 ' −0.32 and se2 ' 0.17 and c2 = 0.95 in
Figure 6(B) where se1 ' 0.17 and se2 ' −0.31. The actual cost ratios becomek=

1 − 0.17
1 − (−0.32)' 1.50 and k =' 0.66.
1.05 − 0.17
1.05 − (−0.31)
17When (λ, k) = (1.1, 1.5), The H-government levies an export tax and the F government gives an export subsidy. On the other hand when (λ, k) = (1.1, 0.66),
the policy is reversed. Notice that the higher-cost firm enjoys the higher subsidy,
which is diﬀerent from the results obtained by de Meza (1986) and Neary (1994).
This contradiction is pointed out and called a "perverse" case by Bandyopadhyay (1997). However, since k > 1 in Figure 6(A) and k < 1 in Figure 6(B), it
can be observed that the firm with the lower actual cost receives higher subsidy,
as a result of the optimal trade policy.

(A) c1 = 1 and c2 = 1.05

(B) c1 = 1 and c2 = 0.95

Figure 6. Asymmetric firms and inelastic demand (λ = 1.1)

3

Dynamics with Elastic Demand

The total dynamic system of the two-stage game consists of the policy dynamic
system with adaptive expectations,⎧⎨ s1 (t + 1) = (1 − α1 )s1 (t) + α1 R1 (s2 (t)),(34)⎩
s2 (t + 1) = (1 − α2 )s2 (t) + α2 R2 (s1 (t)),

where αi is the adjustment coeﬃcient with 0 < αi ≤ 1, and the output dynamic
system with naive expectations,⎧⎨ (1 − λ)x(t + 1) + y(t) = (c1 − s1 (t))(x(t + 1) + y(t))1+λ ,(35)⎩x(t) + (1 − λ)y(t + 1) = (c2 − s2 (t))(x(t) + y(t + 1))1+λ .

Notice that the output dynamic system depends on the variables s1 and s2 ofthe policy dynamic system but not vice versa. In this section we consider the
total dynamics when demand is elastic.

18

We first examine the local stability of the dynamic process of the optimal
policy selection. Its Jacobian matrix is⎛⎞∂RH
1 − α1
α1
⎜
∂s2 ⎟
⎜
⎟
J =⎜
⎟
⎝
⎠
∂RF
α2
1 − α2∂s1

with traceand determinant

trJ = 2 − (α1 + α2 )det J = (1 − α1 )(1 − α2 ) − α1 α2∂RH ∂RF.
∂s2 ∂s1

We can recall that the necessary and suﬃcient conditions for the local asymptotic stability of a two-dimensional system are as follows:
1 − trJ + det J = α1 α2 (1 − γ) > 0,
1 + trJ + det J = 2(2 − (α1 + a2 )) + α1 α2 (1 − γ) > 0,(36)

1 − det J = α1 + α2 − α1 α2 (1 − γ) > 0,with

∂RH ∂RF.
∂s2 ∂s1
The first and second conditions are satisfied if γ < 1 and so is the third condition
if γ > −1. Therefore the stability condition is |γ| < 1. We will next check
whether or not this stability condition is fulfilled in the cases of symmetric andasymmetric firms.
γ=

3.1

Symmetric firms: c1 = c2

The derivatives of the best reply functions evaluated at the steady state aresymmetric if the production costs are identical. Furthermore, in the case ofelastic demand, these are obtained as
0>∂rF (s1 )−5 + 13λ − 11λ2 + 3λ3∂rH (s2 )==> −1 at E e ,∂s2∂s1(λ − 1)2 (13 − 15λ + 4λ2 )and∂rF (s1 )λ2 − 2λ − 1∂rH (s2 )==> 1 at E A .
∂s2
∂s1
λ−1
All derivatives are independent of the value of the common production cost.
Since γ is the product of these derivatives,µ¶2−5 + 13λ − 11λ2 + 3λ3γ=< 1 at E e ,(1 − λ)2 (13 − 15λ + 4λ2 )
19

and
γ=

µ

1 + 2λ − λ2
1−λ¶2

> 1 at E A .

The stability conditions (36) confirm that E e is locally stable and E A is locallyunstable.
By (26), the optimal export subsidies of the two governments are identical.
In consequence, the actual cost ratio is unity when the firms are symmetric.
Returning to the stability condition of the output dynamics, (9), we see thatthe output equilibrium is locally stable, since
¯ µ
¶2 ¯¯
¯
¯ ¯¯
λ
−
1
¯
¯Ψ(λ, k)|k=1 ¯ = ¯−¯ < 1 for 0 < λ < 1.
¯
λ−3 ¯
It is possible to numerically confirm that the stationary points E e in the policy
space and C in the output space are also stable. We summarize these results:

Theorem 3 If the production costs are identical and demand is elastic, then (i)the inner stationary point E e in the policy space is locally asymptotically stable
while the corner point E A is locally unstable; (ii) the output stationary point
associated with E e is locally asymptotically stable.

3.2

Asymmetric firms: c1 6= c2

In this section we consider policy and output dynamics when the firms areasymmetric (c1 6= c2 ) and will see that more exotic phenomena emerge. The
policy dynamic system (34) determines the feedback eﬀect through the export
subsidy policy on the subsequent behavior of the firms and (35) determines
the output adjustment, aiming to arrive at the stationary point. The dynamic
structure of (35) resembles to that of the nonlinear duopoly model studies by
Puu (2003), in which it is shown that the cost diﬀerence is a source of complex
dynamics. The cost diﬀerence also increases nonlinearities involved in (34). One
drawback of introducing the cost diﬀerence is to make the best reply function
more complicated and derivations of analytical solutions very diﬃcult in not
impossible. In such cases, it is a natural way to specify the model and use
numerical simulations to examine how this second-stage game works.
We take c1 = 1, c2 = 1.0567, α1 = α2 = 0.8 and λ = 0.98. In Figure7(A) in which two trajectories, one starting at point a and the other at pointb, are illustrated, the adaptive adjustment process of the traded policy leads
to an asymptotically stable equilibrium point. On the other hand, in Figure
7(B), given the optimal trade policy, the unstable Cournot point gives rise to
chaotic fluctuations. This is a typical example of the situation with λ < 1 inwhich chaotic output dynamics are born although the trade policy is stable.
The actual cost ratio plays a crucial role for the birth of complicated dynamics.
The optimal trade policy is obtained as se1 ' 0.703518, se2 ' −0.64385, and theactual cost ratio isc1 − se1k=' 0.174345.
c2 − se220

Due to (10) and (11), the threshold values of the stability and feasibility areψ 2 (λ)|λ=0.98 ' 0.18124 and φ1 (λ)|λ=0.98 ' 0.17414.
Therefore we have
ψ 2 (λ)|λ=0.98 > k > φ1 (λ)|λ=0.98 .
The first inequality implies instability of the output stationary point and thesecond inequality guarantees the nonnegativity of a trajectory. If k gets closer
to φ1 (λ)|λ=0.98 , then the output fluctuates more aperiodically.

(A) Stable policy equilibrium(B) Chaotic output trajectory

Figure 7. Stable trade policy and chaotic output dynamics
In Figure 8(A) the production cost of firm 2 is increased to c2 = 1.2 withall other parameters kept fixed. As a result, the stationary point of the policy
equilibrium is destabilized. In spite of this instability, the nonlinearities of thesystem prevent the dynamics from diverging and therefore generate bounded
fluctuations around the stationary state (se1 , se2 ). In Figure 8(B) c2 is further
increased to c2 = 1.34 and the degree of elasticity is decreased to 0.96 from
0.98. The policy dynamic system is simulated for 20,000 iterations. The first
5,000 are discarded and the remaining data are plotted in the (s1 , s2 ) space. Itshows the birth of chaotic attoractor. This shows that trajectories aperiodicallyfluctuate in the long-run. Since the output dynamics depends on the policy
dynamics, it also fluctuates aperiodically in the output space. We summarize
these numerical results:
Theorem 4 If the firms are asymmetric and the demand is elastic, then (i)the output dynamics can exhibit chaotic fluctuations even if the policy dynamics
is stable; (ii) the policy dynamics can become chaotic and so can the output
dynamics.

21

(A)

(B)
Figure 8. Birth of chaotic attractors

4

Dynamics with Inelastic Demand

4.1

Symmetric firms: c1 = c2

In the case of inelastic demand, the derivatives evaluated at the inner stationary
point E e have the same form as the ones in the case√of elastic demand. Due
to Assumption 3, it is defined for 1 < λ < λ∗ = 15−8 17 , which is the smaller
root of equation 13 − 15λ + 4λ2 = 0. The derivatives evaluated at the stationary
point are∂rH (s2 )∂rF (s1 )−5 + 13λ − 11λ2 + 3λ3==< −1 at E e .
∂s2∂s1(λ − 1)2 (13 − 15λ + 4λ2 )It follows that the product of the derivatives,γ=µ−5 + 13λ − 11λ2 + 3λ3(1 − λ)2 (13 − 15λ + 4λ2 )¶2> 1 at E e .

Hence E e is locally unstable. Since the product of the derivatives is zero atthe other two equilibria, E a and E b , both points are locally stable. In addition
to the stable stationary points, there exists a stable period-2 cycle along theLb b
diagonal s1 = s2 . The periodic points are (sL
1 , s2 ) and (s1 , s2 ) with
L
L L
b
b
(sb1 , sb2 ) = (R1 (sL
2 ), R2 (s1 )) and (s1 , s2 ) = (R1 (s2 ), R2 (s1 )).

Thus the dynamics system (34) with λ > 1 is characterized by multistability,i.e., coexistence of two attractors and the stable period-2 cycle. In consequence,22

we can construct their basins of attraction to see how the asymptotic behaviorof the trajectories depends on the choice of the initial point.
Figure 9 illustrates the basins of attraction when the policies are naively
adjusted (i.e., α1 = α2 = 1). The sets of point in the policy space such thatthe initial points chosen in the green region converges to the point E a , those in
the blue region evolve to the point E b . Two trajectories, one starting at point
a in the green region and the other starting at point b in the blue region are
shown in Figure 9(A). The red region represents the basin of attraction of theperiod-2 cycle and consists of points which generate trajectories converging to
the period-2 cycle. One stable trajectory is depicted in Figure 9(B) in which
an initial point denoted my c is selected in the upper-right red region. The
trajectory starting at point c repeatedly jumps from one red region to the otherred region and gradually approaches the periodic point on the diagonal. The
following theorem provides the summary of the optimal trade policy:
Theorem 5 If the firms are symmetric and the demand is inelastic, then threeattractors coexist: the two stable fixed points, E a = (sa1 , sa2 ) and E b = (sb1 , sb2 ),and the stable period-2 cycle E cA = (sc1A , sc2A ) and E cB = (sc1B , sc2B ) where
a
L
b
L
b
L
sa1 = sL
1 , s2 = RF (s1 ), s1 = RH (s2 ), s2 = s2

and
(sc1A , sc2A ) = (sa1 , sb2 ) and (sc1B , sc2B ) = (sb1 , sa2 ).

(A) Basins of E a and E b(B) Basin of a period-2 cycle

Figure 9. Basins of attraction when λ > 1
4.1.1

Mixed Trade Policy: E a or E b

We now turn our attention to the output dynamics associated with the trade
policy determined at point E b in which the H-government gives an export subsidy sb1 > 0 to firm 1 and the F -government imposes an export tax sb2 < 0 on
23

firm 2. Since the equilibrium point E b is stable, it is safe to assume that eachfirm receives the optimal value of the trade policy from the beginning of the
dynamic process. Thus the output dynamic system reduces to⎧⎨ (1 − λ)x(t + 1) + y(t) = (c1 − sb1 )(x(t + 1) + y(t))1+λ ,(37)⎩x(t) + (1 − λ)y(t + 1) = (c2 − sb2 )(x(t) + y(t + 1))1+λ .

In the numerical example presented in Figure 10, we take c1 = c2 = 1, sU
1 = c1 ,
L
L
sU
=
c
,
s
=
s
=
−0.854
and
λ
=
1.1.
The
stationary
valuesofthesubsidies2212
are sb1 ' 0.83 and sb2 = −0.854(= sL
2 ). The ratio of the actual costs isk=c1 − se1' 0.09153.
c2 − s22

By (10) and (11), we can obtain the threshold values of the stability and feasibility,ψ 2 (λ)|λ=1.1 ' 0.121695 and φ1 (λ)|λ=1.1 ' 0.09036.
Therefore we have
ψ 2 (λ)|λ=1.1 > k > φ1 (λ)|λ=1.1 .
This inequality condition implies that the stationary point of the outputs is
locally unstable but its trajectory can stay within the feasible (non-negativity)
region. The output trajectory depicted in Figure 10 remains nonnegative for allt ≥ 0 and aperiodically fluctuates around the stationary point C.

Figure 10. Birth of chaotic output dynamics
Depending on the values of the actual cost ratio, the output adjustment
system (37) can generate a wide spectrum of dynamics ranging from stable
dynamics to complex dynamics involving chaos. In Figure 11, a bifurcation
diagram for the output is shown. Each point along the horizontal axis is a value
24

for the lower bound, sL
2 . This is a bifurcation parameter and has the eﬀect ofchanging the actual cost ratio through relation,k=c1 − rH (sL2).
Lc2 − s2

Here the value of sL
2 is increased to −0.8 from −0.855 with 0.0025 increment.
For each value of sL
2 the output equation is simulated for 1200 iterations. The
first 1000 is discarded to eliminate transient changes. The remaining 200 data isLplotted vertically against for sL
2 . As the absolute value of s2 increases, the stationary point is destabilized, bifurcates to a periodic cycle and finally fluctuates
chaotically via a period-doubling cascade. Under the symmetric assumption
c1 = c2 , we can examine the output dynamics associated with point E a in thesame way. We summarize the results as follows:
Theorem 6 Assume that the firms are symmetric, demand is inelastic andthe governments takes the mixed policy, either E a or E b . The output dynamics
exhibits complicated dynamics if its stationary point is locally unstable.

Figure 11. A bifurcation diagram for the output dynamic system (37).
4.1.2

Periodic Trade Policy

The shape of the basin of attraction is sensitive to the value of the adjustment
coeﬃcient αi . Figure 12 illustrates two basins of attraction with two diﬀerent
values of αi and indicates that the red regions become smaller with decreasing25

value of αi . We will reveal the mechanism which makes the red regions shrink.

(A) α1 = α2 = 0.9(B) α1 = α2 = 0.86

Figure 12. Basins of attraction with two diferent values of αi
Any trajectory starting from a point in the red region converges to theperiod-2 cycle with its periodic points belonging to the diagonal. Hence one
way to consider appearance or disappearance of a period-2 cycle is to restrict
the dynamic system to the diagonal of the policy space by assuming that thetwo firms have the same adjustment coeﬃcients, α1 = α2 = α. This assumption
together with the assumption of the identical production costs imply that thetwo firms behave identically if an initial point is selected on the diagonal. So
their dynamic behavior can be described by the following one-dimensional map,
which describes a representative firm:s(t + 1) = ϕ(s(t)) = (1 − α)s(t) + αR(s(t))
with R(s) = R1 (s) = R2 (s). The fixed point of ϕ(s) isse =(1 − λ)λc2(3 − 2λ)and the derivative of ϕ(s) at the fixed point becomes
¶
µ
dϕ(s)
1
2α(3λ − 5)=.
2(1 − α) +ds |s=se2
13 − 15λ + 4λ2
It is possible to show that the√ derivative is less than unity for 0 < α ≤ 1 and1 < λ ≤ λ∗ where λ∗ = (15− 17)/8. It then follows that the stability condition
for the fixed point is
dϕ(s)
−1 <
ds |s=seor¢
9 − 9λ + 2λ2 ¡ S
0>
2 α −α13 − 15λ + 4λ
26

with

13 − 15λ + 4λ2.
9 − 9λ + 2λ2
Hence we arrive at the following stability conditions:αS =If α < αS , then se is locally stable.
To examine the birth of a period-2 cycle, we define the second iteratedfunction of ϕ(s) by ϕ(ϕ(s)) and introduces a new function φ(s) = s − ϕ(ϕ(s)).
Figure 13 in which λ = 1.1 illustrates four graphs of φ(s) with α = 0.89,
α = 0.86, α = 0.84 and α = 0.8478. The graphs with the first three valuesof α are depicted as solid curves and the one with α = 0.8478 as the bold
curve. The intersections with the horizontal line are the fixed points of ϕ(ϕ(s)).
Since se is the fixed point of ϕ(s), it is also a fixed point of ϕ(ϕ(x)) and all
curves pass through this point. For α = 0.89, the φ(s) curve has a negative
slope at se because αS ' 0.88156 < 0.89 implies the instability of se . It has
an N -shaped curve and its positive sloping parts cross the horizontal line togive rise to a periodic points of a stable period-2 cycle. For α = 0.86 < αS ,se is locally stable. The φ(s) curve crosses the horizontal line five times andthus generates two period-2 cycles, one inner unstable cycle and the other outerstable cycle. For α = 0.84, se is locally stable but the φ(s) curve intersects thehorizontal line only once at se . This implies that there exist no period-2 cycle.
The threshold value of α∗ ' 0.8478 distinguishes the second case (i.e., emergenceof two periodic cycles) from the third case (no periodic cycle). The φ(s) curvewith α∗ is depicted as bold and touches the positive part of the horizontal linefrom above and the negative part from below. In other word, the inner cycle
coincides with the outer cycle. Therefore two distinct periodic cycles emerge forα > α∗ and no cycle emerges for α < α∗ . For α < α∗ the red region disappears
and the initial diﬀerence in the policy determines the equilibrium trade policy.
Indeed any trajectory starting at a point with s1 (0) > s2 (0) converges to theequilibrium point E b while any trajectory starting a point with s1 (0) < s2 (0)27approaches the equilibrium point E a .

Figure 13. Emergence of a period-2 cycle
Theorem 7 Suppose that the production costs are identical and the demandis inelastic. (i) The stationary point is locally unstable and one period-2 cycle
exists for 1 ≥ α ≥ αS ; (ii) The stationary point is locally stable and two period2 cycles exist for αS > α ≥ α∗ ; (iii) The stationary point is locally stable andno periodic cycles exist for α < α∗ where the critical value and the equilibrium
value of the adjustment coeﬃcient are
αS ' 0.8816 and α∗ ' 0.8478 if λ = 1.1.
Since the output dynamics depends on the policy dynamics but not viceversa, we can be fairly certain that the output dynamics gives rise to a period-2
cycle which is synchronized with the period-2 cycle of trade policy.

4.2

Asymmetric Firms: c1 6= c2

The cost diﬀerence does not change the qualitative properties of the dynamics
when the demand is inelastic. The policy space is divided into three parts,
each of which is a basin of attraction of a stationary state if the adjustment
coeﬃcients are close to unity. A period-2 cycle and its basin disappear if thecoeﬃcients become much smaller than unity. The cost diﬀerence and the valueof the adjustment coeﬃcient, however, quantitatively aﬀects the shape of thebasin. We fix c1 = 1, λ = 1.1, α1 = α2 = 0.95 and take c2 = 1.05 in Figure
14(A) and c2 = 0.95 in Figure 14(B). We have already examined the eﬀect on
the determination of the optimal trade policy caused by the cost asymmetryin Figure 7 in which the same values of parameters are taken. It is still truethat the higher-cost firm can enjoys the higher subsidy at point E a in Figure
14(A) and at point E b in Figure 14(B). On the other hand, the lower-cost firm
receives the higher subsidy at point E b in Figure 14(A) and at point Ea in28

Figure 14(B). Comparing Figure 9(A) and Figure 14(A) reveals that the basinof E a becomes smaller when the cost of firm 2 is larger and so does the basin of
E b when the cost of firm 1 is larger. Then it becomes probable that the lowercost firm receives higher subsidy as the cost diﬀerence becomes larger. It isalso numerically confirmed that the basin of the period-2 cycle disappears when
the adjustment coeﬃcient becomes smaller. Since the best replies of the firms
become complicated with cost diﬀerence, it is not easy to reveal analytically themechanism that causes disappearance of the period-2 cycle.

(A) c2 = 1.05

(B) c2 = 0.95

Figure14. Distorted basins of attraction with asymmetric firms

5Concluding Remarks

In this paper we assume that the price function is hyperbolic and constructa three-country model with two active governments and two firms to consider
dynamic behavior of the sequential subsidy game in which the governments
determine their trade policy and the firms determine their optimal outputs. We
first deal with the determination of the governments’ optimal trade policy that
depends on the actual cost ratio and strategic characteristics of the outputs.
Our dynamic results are summarized as follows:(1) When the production costs are identical, a trade policy and the corresponding optimal output are stable if the demand is elastic while multistability(i.e., coexistence of multiple attractors) and complex dynamics of output
occur if the demand is inelastic.
(2) When the production costs are diﬀerent, a stable trade policy can induce
chaotic output fluctuations regardless of demand elasticity.
(3) When the production costs are diﬀerent, the trade policy can be chaotic
and so does the output if the demand is elastic while multistability andchaotic output dynamics occur if the demand is inelastic.

29

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