Dynamics in International Subsidy Games with
Unit-elasitic Demand∗
Akio Matsumoto†Chuo University

Ferenc Szidarovszky‡
University of Arizona

Abstract
In this paper, we construct a three-country model with two governments and two firms and consider dynamic behavior of the sequential subsidy game in which the governments determine their optimal trade policies
and then the firms determine their optimal outputs. We first show theexistence of an optimal trade policy under realistic conditions. In the caseof symmetric firms, the governments adopt periodic mixed trade policy(i.e., one government gives subsidy and the other levies tax in one period
and then the governments interchange their policies in the next period) if
the adjustment is naive, and the governments adopt a stable mixed policy
if adaptive. In the case of asymmetric firms, a firm receives subsidy if its
cost is lower and pays tax if higher. If the Cournot output point under
the optimal subsidy is locally stable, then its dynamics can be periodic
which is synchronized with the periodic trade policy. If it is locally unstable, then complex dynamics involving chaos emerges regardless of thecost diﬀerence.

1

Introduction

Markets become imperfectly competitive due to many factors such as the small
number of firms, the diﬀerentiated goods, the scale of economics, etc. In suchan imperfectly competitive international market, the governments may be motivated to introduce trade policies like tariﬀ, export subsidy and tax to increase
national welfare of their countries. It has been demonstrated that an increase ina domestic export subsidy raises the domestic firm’s output and its profit when
the firms compete in a Cournot way (Brander and Spencer (1985)). It has been
also demonstrated that an export tax can be optimal when the firms compete∗ 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 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

in a Bertrand way (Eaton and Grossman (1986)). It is now well-known thatthe source of this sharp contrast comes from the diﬀerence in the assumption
on the firms’ strategic behavior (that is, the goods are strategic substitutes or
strategic complements). It is also well-known that this behavioral diﬀerence
relates to the curvature and the elasticity of the demand function. Recently,
constructing a simplified version of the three-country model with two firms andtwo governments, Bandyopadhyay (1997) shows, among others, the following
clear-cut results when the demand is unit-elastic:(1) When the foreign government is passive, the optimal trade policy of theactive domestic government is free trade if the production costs of the two
firms are identical, an export subsidy if the home firm has lower cost andan export tax otherwise.
(2) When both governments are active, a continuum of policy equilibria exist if
the production costs are identical and no policy equilibrium exists if they
are diﬀerent.
As a consequence of the second result, no dynamic consideration has been
provided yet. In particular, it is not known yet how the optimal policy as well as
the optimal outputs change over time and what kinds of changes might occur.
The main purpose of this paper is to consider dynamics of the optimal trade
policy and the associated optimal outputs under unit-elastic demand and toshow that rich dynamics can be born when natural constraints are imposed on
the government’s policy selections. This paper complements Bandyopadhyay(1997) from a dynamic point of view. It is a continuation of Matsumoto andSerizawa (2007) who focus mainly on the comparative statistic analysis of thesimilar model (i.e., the eﬀects on the optimal outputs caused by a change inthe trade policy of the domestic government). The dynamic model of outputs
to be considered in this paper resembles nonlinear dynamic duopoly models,
which have been extensively studies for the last twenty years. Comprehensive
summary of the earlier work has been presented in Puu (2003). More recent
developments on this field are given in Bischi, et al. (2009). This paper also
aims to apply the theoretical results obtained there to the dynamic analysis in
the framework of international economics.
The paper is organized as follows. Section 2 presents a variant of the threecountry model in which both governments are active. Section 3 considers policy
dynamics and Section 4 analyzes output dynamics with the optimal trade policy.
Section 5 gives concluding remarks.

2

Model

The model presented below is a variant of the three-country model. There
are two countries with one firm in each of them, and these firms export theirproduct to a third country. The outputs of the firms are denoted by x and y, andconstant marginal costs of the two firms are denoted by c1 and c2 , respectively.
2Competition in the third country is modeled through a two-stage game. At
the first stage, the governments hosting their firms choose subsidy rates, si for
i = 1, 2, in order to maximize their welfare, taking the optimal behavior of thefirms as given. At the second stage, the firms employ the quantity competition ina Cournot way and choose profit maximizing outputs, taking their governments’
trade policies as given. Optimal subsides and optimal outputs are backwardly
determined. In particular, we solve the profit maximization problems of thefirms, given the levels of the subsidy in Section 2.1, then examine the welfare
maximization problems of the governments, given the optimal behavior of thefirms and determine the optimal levels of the subsidy in Section 2.2. In orderto get a complete description of the dynamics of the subsidy game in the latterpart of the paper, we will specify the best reply functions of the firms and thoseof the governments in this section.

2.1

Profit Maximization

Let the inverse demand function be unit-elastic,
P =

1
,
Q

where Q is the total output, Q = x + y.1 At the second stage in which thegovernments’ subsidies are given, firm x and firm y choose outputs to maximize
their profits defined by
π1 = (P − (c1 − s1 ))x,and
π 2 = (P − (c2 − s2 ))y.
The first-order conditions of the profit maximization are given by∂π 1y− cx = 0,=∂x(x + y)2
and

x
∂π 2
− cy = 0,=∂y(x + y)2

where cx = c1 − s1 and cy = c2 − s2 for notational simplicity.2 We call the
production cost including the subsidy an actual cost. Although we will formallyshow later that the actual costs are non-negative, we suppose for the time beingthat subsidies are given as cx > 0 and cy > 0. From the first-order conditions,the explicit forms of the firms’ best reply functions are derived asryr̄1 (y) =−y(1)cx
1 See Matsumoto and Szidarovszky (2009) that studies the same model with diﬀerent demand, P = Q−λ and λ 6= 1.
2 It can be checked that the second-order conditions are satisfied for any x and y that solve
the first-order conditions.

3andr̄2 (x) =

r

x
− x.
cy

(2)

An intersection of the best reply functions determines a Cournot equilibrium atwhich we can solve for the output quantities:xC (s1 , s2 ) =cy(cx + cy )2(3)

y C (s1 , s2 ) =cx,(cx + cy )2

(4)

and

where superscript C is attached to variables associated with the Cournot point.3
The Cournot outputs in (3) and (4) are substituted into the profit functions to
obtain the Cournot profits:
µ
¶2
cy
C
(5)
π1 (s1 , s2 ) =cx + cyandπC
2 (s1 , s2 ) =µcx
cx + cy

¶2

.

(6)

Dividing (3) by (4) and (5) by (6) yields, after arranging terms, the ratios of
outputs and profits,cyxC=
R 1 according to cy R cx ,
C
y
cx
and
πC
x
=
πC
y

µ C ¶2x
R 1 according to xC R y C .
yC

These inequalities imply the following results on the optimal behavior of thefirms: The firm with the lower actual cost produces more output and earns
more profit than the firm with the higher actual cost.

2.2

Welfare Maximization

At the first stage of the sequential game, the governments determine the optimallevels of the subsidy so as to maximize the national welfare defined byC
W1 (s1 , s2 ) = π C
1 (s1 , s2 ) − s1 x (s1 , s2 ),(7)C
W2 (s2 , s1 ) = π C
2 (s2 , s1 ) − s2 y (s2 , s1 ).

(8)

and
3 Since r̄ (y) and r̄ (x) take mound-shaped curves starting at the origin, the curves inter1
2
sects twice at (0, 0) and (xC , yC ). The former is the trivial equilibrium point and the latter isthe non-trivial equilibrium point. Our concern is on the non-trivial point and thus no further
considerations are given to the trivial equilibrium point.

4We derive the specific forms of the best reply functions of the governments andconsider their characteristics in the policy space. Substituting xC , y C , QC =xC + y C and P C = (QC )−1 into (7) and (8) yields the explicit forms of thewelfare functions of the governments,
W1 (s1 , s2 ) =(cy − s1 ) cy,
(cx + cy )2(9)W2 (s2 , s1 ) =(cx − s2 ) cx.
(cx + cy )2

(10)

and

The government of country 1 maximizes W1 (s1 , s2 ) with respect to s1 and thegovernment of country 2 maximizes W2 (s2 , s1 ) with respect to s2 . We can solve
the first-order conditions to obtain the best reply functions:r1 (s2 ) = −s2 + (c2 − c1 ) and r2 (s1 ) = −s1 + (c1 − c2 ),(11)where the second-order conditions are satisfied. These functions are essentiallythe same as those derived by Bandyopadhyay (1997). It is apparent that there
is a continuum of optimal subsidies s1 + s2 = 0 for symmetric firms (c1 = c2 )and no equilibrium exists for asymmetric firms (c1 6= c2 ).
To avoid the indeterminacy of the optimal policy equilibrium in the caseof asymmetric firms, we impose the following external upper and lower boundconstraints on the levels of the optimal policy, si , taking account of the factthat the governments behave with control. The first constraint reflects the factthat the governments have the upper bound of the subsidy, due to their budget
constraints. The second constraint takes account of the fact that the government
does not levy such a strong export tax that might result in its firm to exit themarket. Intuitively speaking, in choosing their policies, the governments neither
take care of all the production costs nor take all of the profits.
U
U
Assumption 1. sLi ≤ si ≤ si for i = 1, 2 where si is the upper bound of theULsubsidy level defined by si = ci and si < 0 is the lower bound of thesubsidy level, which shows the upper bound of the export tax.

Under Assumption 1, the best reply function of the government of country
1 becomes piecewise linear with three segments:⎧ U
s2 < su2 ,⎨ s1
r1 (s2 ) su2 ≤ s2 ≤ s`2 ,⎩ L
s2 > s`2 ,s1`Lwhere su2 and s`2 are defined by r1 (su2 ) = sU
1 and r1 (s2 ) = s1 , respectively. In
the same way, the best reply function of the government of country 2 is derived
to be piecewise-linear with three segments:⎧ U
s1 < su1 ,⎨ s2
r2 (s1 ) su1 ≤ s1 ≤ s`1 ,⎩ L
s1 > s`1 ,
s2

5

`
L
where su1 and s`1 > 0 are defined by r2 (su1 ) = sU
2 and r2 (s1 ) = s2 , respectively.
An intersection of these modified best reply functions is a Nash equilibriumof the trade policy, (se1 , se2 ). First of all, we should discuss the determination ofu
the optimal trade policy in the case where ci < cj and sLj ≤ sj . By definitionof the piecewise linear best replies, it is clear that the optimal subsidy policy ofUfirm i is sei = sU
i . From (3), (4) and Assumption 1 (i.e., si = ci ), the optimalLuoutput of firm 2 is zero if c1 < c2 and s2 ≤ s2 whereas the optimal output
uof firm 1 is zero if c2 < c1 and sL
1 ≤ s1 . In both cases one of the firms will
export nothing to the third country and the competition in the third country
will be terminated. To confine our attention to the third-country model withactive competition, we assume that suj and sL
j are given such that the following
inequality holds.

Assumption 2.

usL
j > sj for j = 1, 2.

u`Notice that suj = cj − 2ci and s`j = cj − ci − sL
i , so clearly sj < sj for bothufirms. Assumption 2 requires that for j = 1, 2, sL
j > sj . In order to guaranteeLthe existence of negative sj bounds, we make the additional assumption:

Assumption 3.

cj < 2ci for j = 1, 2 and i 6= j.

Without losing generality, we can assume that in the case of asymmetricL
firms, ci < cj . First we show that s`i < sU
i . Since sj > cj − 2ci > cj − 2cj = −cj ,we haveUs`i = ci − cj − sL
j < ci − cj − (−cj ) = ci = si .
`However no such comparison can be made between sL
i and si . In order to guarL`
antee that si < si , we make the following assumption:

Assumption 4.

LsL
i + sj < ci − cj .

Notice that this assumption holds if the marginal costs ci and cj are close
to each other. The best reply functions are shown in Figure 1, and from it wecan conclude that the unique equilibrium iseLsei = cj − ci − sL
j and sj = sj

(12)

Hence we have the following:
Theorem 1 Under Assumptions 1,2,3 and 4, if the firms are symmetric (i.e.,c1 = c2 ), then there are infinitely many equilibria on the line si = −sj whereas
if the firms are asymmetric with ci < cj , then sei = −sL
j + (cj − ci ) > 0 and
sej = sL<0.
j6

Determination of the optimal policy under ci < cj3

Dynamics Analysis of Trade Policy

Theorem 1 guarantees the existence of a unique optimal trade policy. In thissection, we consider the dynamic behavior of the trade policy in the policy space(s1 , s2 ). To this end, we assume the following discrete time dynamic process ofthe policy selection:⎧ 0⎨ s1 = (1 − α1 )s1 + α1 R1 (s2 ),(13)⎩ 0
s2 = (1 − α2 )s2 + α2 R2 (s1 ),where 0 denotes the unit-time advancement operator, αi is the adjustment coeﬃcient with 0 < αi ≤ 1 and R1 (s2 ) and R2 (s1 ) are the best reply functions
U
L U
restricted to their intervals, [sL
2 , s2 ] and [s1 , s1 ],⎧ L⎨ s1 for s`2 ≤ s2 ≤ sU
2,
R1 (s2 ) =⎩`r1 (s2 ) for sL
2 ≤ s2 < s2 ,andR2 (s1 ) =

3.1

⎧ L
⎨ s2 for s`1 ≤ s1 ≤ sU
1,⎩`
r2 (s1 ) for sL
1 ≤ s1 < s1 ,Symmetric firms: c1 = c2

In the case of identical costs, we first perform some numerical simulations to
examine the dynamic behavior of the governments, second, confirm analyticallythat the numerical results are robust, and finally summarize these results in
7

Theorems 2 and 3. The numerical simulations are presented in Figure 2 where
the policy is naively adjusted (i.e., αi = 1). The other case is given in Figure 3
where the policy is adaptively adjusted (i.e., αi < 1).
U
L U
The feasible policy space is defined by the rectangle S = [sL
2 , s2 ] × [s1 , s1 ],which is divided into distinctive four parts by the horizontal and vertical lines,
s1 = s`1 and s2 = s`2 ,SI = {(s1 , s2 ) ∈ S | s`1 ≤ s1 and s`2 ≤ s2 },SII = {(s1 , s2 ) ∈ S | s1 < s`1 and s`2 ≤ s2 },SIII = {(s1 , s2 ) ∈ S | s`1 ≤ s1 and s2 < s`2 },SIV = {(s1 , s2 ) ∈ S | s1 < s`1 and s2 < s`2 }.
In Figures 2(A) and 3(A), we select three initial points denoted as I1 , I2 andI3 in SI , SII and SIII , respectively and depict the three trajectories startingfrom these points. In Figures 2(B) and 3(B), which are enlargements of theregion SIV , we also select three initial points denoted as i1 , i2 and i3 inside
SIV and depict three trajectories starting from these points. Simulations inFigure 2(A) indicate that the trajectories converge to period-2 cycles when the
trade policies are naively adjusted. The two periodic points are symmetric withrespect to the line s1 + s2 = 0. On the other hand, the simulations shown in
Figure 3(A) indicate that the trajectories converge to stationary points on theline s1 + s2 = 0 when the trade policies are adaptively adjusted.
The first result on policy dynamics is summarized as follows:
Theorem 2 If the firms are symmetric, then the naively adjusted process ofthe export trade policy (i.e., (13) with α1 = α2 = 1) gives rise to infinitely
many stable period-2 cycles, and a trajectory starting from a point other than astationary point converges to one of these cycles.
Proof. We prove this statement with four steps. (I): It can be seen thatL
R1 (s2 ) = sL
1 and R2 (s1 ) = s2 for all (s1 , s2 ) ∈ SI . By the identical cost asL
`
`
`
L
`
L
sumption, R1 (s2 ) = s1 and R2 (sL
1 ) = s2 whereas R1 (s2 ) = s1 and R2 (s1 ) = s2``
by the definitions of s2 and s1 . Thus any trajectory starting at a point insideL` `S1 converges to the period-2 cycle with periodic points, (sL
1 , s2 ) and (s1 , s2 ).
The trajectory with the initial point I1 in Figure 3(A) is an example of thisIIcase. (II): Next take an initial point (sII
1 , s2 ) ∈ SII . Then the naive adjustIIII
II
ment process conveys the point to R1 (s2 ) = sL
1 and R2 (s1 ) = −s2 and then
II
II
L
`
II
R1 (−s2 ) = s1 and R2 (s1 ) = s2 , that are bounced back to the point (sL
1 , −s2 ).
II IIThus any trajectory starting at a point (s1 , s2 ) ∈ SII converges to the periodIIII `2 cycle with periodic points (sL
1 , −s2 ) and (s1 , s2 ). The trajectory with theinitial point I2 is an example of this case. (III): In the same way, we can show
IIIthat a trajectory starting at a point (sIII
1 , s2 ) ∈ SIII converges to the period-2
III L
cycle with periodic points, (−s2 , s2 ) and (s`1 , sIII
2 ). The trajectory with theinitial point I3 is an example of this case. (IV): We finally consider periodic
8

behavior in Figure 3(B) where the initial points are selected inside of SIV . Since
the adjustment process in SIV is given by
s01 = −s2 and s02 = s1 ,IV
the governments expect the symmetric point (−sIV
1 , −s2 ) with respect to theIVIV
s1 +s2 = 0 locus for any initial point (sIV,s).
Taking(−sIV
12
1 , −s2 ) as given,IVthe government expect (sIV
,
s
)
in
the
next
time
period.
Thus
any point
1
2
IV
IV
IV
(sIV
,
s
)
∈
S
and
its
symmetric
point
(−s
,
−s
)
∈
S
are
period-2
IV
IV
1
2
1
2
points. Three period-2 cycles depicted in Figure 2(B) are examples of this case.

(A) Convergence to SIV(B) Period-2 cycles in SIV

Figure 2. Coexistence of period-2 cycles under naive expectation
We say that the governments take the pure policy if s1 s2 > 0, the mixed
policy if s1 s2 < 0, and the one-side free trade policy if s1 s2 = 0 and si 6= 0
for i = 1, 2. Furthermore, we say that the governments take a pure subsidy
policy if s1 > 0 and s2 > 0 and a pure tax policy if s1 < 0 and s2 < 0. In themixed policy, one government pays subsidy and the other government charges
tax. Figures 2(A) and 2(B) are divided into four rectangles by the horizontal
and vertical axes. The rectangles on the top-right and the bottom-left represent
the set of points which generate pure policy (i.e., the pure subsidy policy andthe pure tax policy). On the other hand the rectangles on the top-left andon the bottom-right represent the set of points which generate mixed policy.
If a point is on the either axis, it represents the one-side free trade in which
one government gives no subsidy and charges no tax and the other government
either gives subsidy or charges tax.
A
Let us denote the period-2 points of the trade policy by S A = (sA
1 , s2 ) andB
B B
S = (s1 , s2 ). The period-2 cycle means that if the governments expect S A ,then S B is realized; and if the governments expect S B in the next period, thenS A is realized. Let us take the case of the pure trade policy. If both governments
9expect that their competitors take the same policy, for example export subsidies
AsA
1 > 0 and s2 > 0, then in the next period the process becomes the opposite
Bpure policy, that is, export taxes sB
1 < 0 and s2 < 0. So the governments
alternate between the pure subsidy policy and the pure tax policy. In contrast
to this, if both governments expect that their competitors take the mixed policy,
for example, government 1 adopts the subsidy policy sA
1 > 0 while government
2 takes the tax policy sA
<
0,
then,
in
the
next
time
periodthe process becomes2
AA
the subsidy with sB=−sforgovernment
1
and
the
tax
with sB12
2 = −s1 forgovernment 2. The governments therefore alternate between the mixed policies
in which the values and the signs of the subsidy and the tax are interchanged.
We summarize these results as a corollary of Theorem 2:Corollary 1 The trade policy has an initial point dependency: (1) If bothgovernments expect that their competitors adopt the subsidy policy, then
the tax policy is realized, and vice versa. (2) If a mixed policy is expected,
then the opposite mixed policy is realized where the realized point of thesubsidy or the tax is mirror image of the expected point with respect to the-45 ◦ degree line.
The optimal outputs associated to these periodic points are obtained bysubstituting the periodic points into (3) and (4): for k = A, B,
xCk =c2 − sk2
k((c1 − s1 ) + (c2 − sk2 ))2and y Ck =c1 − sk1.
k((c1 − s1 ) + (c2 − sk2 ))2(14)

We now assume that the policy is adaptively adjusted (i.e., αi < 1) andthe adjustment coeﬃcients are the same (i.e., α1 = α2 = α) for the sake ofanalytical simplicity. As can be seen in Figure 3(A), any trajectory with an
initial point inside SI ∪ SII ∪ SIII sooner or latter enters SIV . It is therefore
suﬃcient for our purpose to consider the dynamics observed within SIV . Our
second result on policy dynamics is summarized as follows:
Theorem 3 If the firms are symmetric, then the symmetric adaptive adjustment process of the export trade policy (i.e., (13) with α1 = α2 < 1) is stable
and converges to a point on the line s1 + s2 = 0.
Proof. Let us start with an initial point (s01 , s02 ) ∈ SIV . The optimal policy at
the next period is determined by the adaptively adjusted process,0

s1 = (1 − α)s01 + α(−s02 ),0

s2 = (1 − α)s02 + α(−s01 ).
The line passing through these two points, (s01 , s02 ) and (s01 , s02 ), is written ass2 = as1 + b where the slope a and the vertical intercept b area=s02 − s02
= 1 and b = s02 − s01 .
s01 − s01
10

It is clear that the adaptive process maps the optimal policy (s01 , s02 ) to a pointon the s2 = as1 + b locus. That is, the trajectories of the optimal policies arecontrolled by0
s1 = (1 − α)s1 + α(−s1 − b),0s2 = (1 − α)s2 + α(−s2 + b).
The adjusted processes are independent and governed by 1D diﬀerence equations
with slopes less than unity in absolute value,¯ 0¯¯ ds ¯¯ i¯¯¯ = |1 − 2α| < 1.
¯ dsi ¯Hence the adjustment process is stable and a trajectory converges oscillating to
the stationary state associated with the initial point (s01 , s02 ),s0 − s01
s0 − s01
se1 = − 2and se2 = 222which is the intersection of the s2 = −s1 curve and the s2 = as1 + b curve. It
is clear that se1 R 0 and se2 Q 0 if s01 R 0 and s02 Q 0.

(A) Convergence to SIV(B) Stable points in SIV

Figure 3. Coexistence of stable stationary points under adaptive expectation
The stationary point (se1 , se2 ) is on the line s1 + s2 = 0. Figure 3(B) showsthat the trajectory starting from point i1 converges to the origin, se1 = se2 = 0,α
the trajectory starting from point i2 converges to point α with sα
1 < 0, s2 > 0αα
and s1 = −s2 , and the trajectory starting from i3 converges to point β withsβ1 > 0, sβ2 < 0 and sβ1 = −sβ2 . This observation leads us to Corollary 2.
Corollary 2 The optimal trade policy has an initial point dependency: if bothgovernments start with the same initial expectations, then the free trade is11materialized (i.e., se1 = se2 = 0); If they start with diﬀerent initial expectations (i.e., s01 6= s02 ), then the symmetric mixed trade policy is materialized:
s0 − s01
se1 = − 2and se2 = −se1 .
2
The corresponding stationary output values are obtained by substituting se1
and se2 into (3) and (4),
xC ==

3.2

c2 − se2e((c1 − s1 ) + (c2 − se2 ))2and y C =c1 − se1.
e((c1 − s1 ) + (c2 − se2 ))2(15)

Asymmetric firms: c1 6= c2

It is clear from Figure 1 that the optimal policy is stable in the policy space if
the firms are asymmetric, regardless of whether the policy is naively or adaptively adjusted. It is also clear that the optimal trade policy is mixed, which issummarized as follows:
Theorem 4 If the firms are asymmetric, then the firm with lower production
cost receives an export subsidy and the firm with higher production cost pays anexport tax.
The optimal output values are determined by substituting the optimal subsidies (12) into relations (3) and (4).

4Dynamic Analysis of Output

In considering the output dynamics in the international subsidy game, we lag
the output variables in (1) and (2) and construct the output adjustment processwith adaptive expectations:
¶
µr
⎧
y
⎪
0
⎪
x
=
(1
−
β
)x
+
β
−
y
,
⎪
1
1
⎪
c1 − s1
⎨
(16)
¶
µr
⎪
⎪
x
⎪
⎪
−x ,
⎩ y 0 = (1 − β 2 )y + β 2c2 − s2

where β i is the adjustment coeﬃcient satisfying 0 < β i ≤ 1, and s1 and s2 aregoverned by (13). As mentioned in the Introduction, the dynamic structure ofsystem (16) resembles that of the nonlinear Cournot models extensively studied
by Puu (2003) and Bischi et al. (2009) in which diﬀerent subjects such as the
emergence of complex dynamics involving chaos, multistability, the structure of
the basin of attraction, delay dynamics, etc., are discussed. We skip the detailed
examinations of system (16) and will apply these results to our dynamic analysis.

12

4.1

Symmetric firms: c1 = c2

As a benchmark, we take β 1 = β 2 = 1 and consider the output dynamics undernaive expectation. Let us denote the actual cost ratio byk=c1 − s1.
c2 − s2

It is already shown in Puu (2003) that loss of stability occurs when the actual
cost ratio satisfies the following equation,4(k − 1)2
=1
(17)
4k
√
where the smaller solution is 3 − 2 2(' 0.172) and the larger solution is 3 +√˙ It follows that if the actual cost ratio stays within the interval
2 2(' 5.828).
bounded by the smaller and larger solutions, then the Cournot point is stable.
In the same way if the actual cost ratio falls outside the interval, it becomes
locally unstable. We assume the stability of the Cournot point and examine theeﬀects caused by the policy on output dynamics for a while. It is also shown in
Puu (2003) that the dynamics is symmetric with respect to k = 1. In order to
get new results, we confine our consideration to the case of c1 < c2 and assume
that c1 = 1 throughout the analysis for the sake of analytical convenience. It
can be checked that the nonnegativity of the output trajectories is guaranteed
when the actual cost ratio is at least 4/25(= 0.16).
If there is no policy lag in the sense that the firms receive the subsidies fromthe governments without any time delays, and the policy is naively adjusted,
then the dynamics of the outputs and the subsidies are controlled by the dynamic
equations
⎧
r
y
⎪
0
⎪
x
=
− y,
⎪
⎪
c1 − s1
⎪
⎪
⎪
⎪
⎪
⎪
r
⎪
⎪
x
⎨ 0
− x,
y =
(18)
c2 − s2⎪⎪⎪⎪⎪⎪
s01 = −s2 ,
⎪
⎪
⎪
⎪
⎪
⎪
⎩ 0
s2 = −s1 ,`where the domain of the policy dynamics is restricted to the rectangle, [sL
1 , s1 ]×L `5[s2 , s2 ] for analytical simplicity. According to Theorem 2, the optimal trade
A
B
B
policy oscillates between two values, S A = (sA= (sB
1 , s2 ) and S
1 , s2 ) whereA
B
A
B
s1 = −s2 and s2 = −s1 . We have already solved for the output quantities at

4 This equation is obtained by setting the product of the derivatives of the best reply
functions, r̄1 (y) and r̄2 (x), evaluated at the Cournot point equal to −1.
5 As can be seen in Figure 2(A), any trajectory starting at a point outside S
IV will enter
SIV after several iterations.

13each periodic point given in (14). Indeed, the Cournot outputs at point S A arexCA =c2 − sA
2
A
2((c1 − s1 ) + (c2 − sA
2 ))and y CA =c1 − sA
1
A
2((c1 − s1 ) + (c2 − sA
2 ))and the Cournot outputs at point S B arexCB =c2 − sB
c1 − sB
2
1
CB
and
y
=
.
B 2
B 2
((c1 − sB((c1 − sB
1 ) + (c2 − s2 ))1 ) + (c2 − s2 ))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 periodic
cycle when the trade policy has a period-2 cycle. We specify the parametervalues as c1 = c2 = 1, S A = (0.64, −0.4) and S B = (0.4, −0.64) and perform
simulations. First of all, it can be pointed out that the output equilibria, C A =(xCA , y CA ) and C B = (xCB , y CB ), are locally asymptotically stable under these
specifications
if the trade policy is fixed since the cost ratios are greater than√
3 − 2 2,c1 − sA
c1 − sB1
1' 0.366 and kB =' 0.257.
kA =A
c2 − s2c2 − sB
2

However, the trade policy is not fixed, it is switched from one periodic point
to the other in every period. Figure 4 reveals that the output dynamics isrepresented by a period-2 cycle, which is synchronized with the period-2 cycle ofthe optimal subsidy. Figure 4(A) shows a return map. The best reply functionsof firm 2 are illustrated as mound-shaped curves and the inner curve is shiftedto the outer curve when the policy is switched from S A to S B . In the same way,
the best reply functions of firm 1 are illustrated as two upward sloping curves
and the shift from the left curve to the right is caused by the policy switchingfrom S A to S B . Here C a and C b are the two periodic points of the output cycle.
Figure 4(B) depicts the time trajectory of output y. Contrary to our intuition,
the periodic points of the output cycle are not the Cournot points denoted by
C A and C B in Figure 4(A). The reason is that dynamic equations of the outputs
are switched from (x0 , y 0 ) = (r̄1A (y), r̄2A (x)) with S A to (x0 , y 0 ) = (r̄1B (y), r̄2B (x))
with S B at every iteration step where
r
r
y
x
A
r̄1A (y) =
−
y
and
r̄
(x)
=
−x
2
c1 − sA
c
−
sA
2
1
2
and
r̄1B (y) =ry− y and r̄2B (x) =c1 − sB
1

r

x
− x.
c2 − sB2

The periodic points C a = (xa , y a ) and C b = (xb , y b ) are the fixed points of thecomposite functions of r̄iA and r̄iB for i = 1, 2,
xa = r̄1A (r̄2B (xa )), y a = r̄2A (r̄1B (y a )), xb = r̄1B (r̄2A (xb )) and y b = r̄2B (r̄1A (y b )).
We summarize these results as follows:14

Theorem 5 If the firms are symmetric, the trade policies and the outputs arenaively adjusted and there is no policy lag, then the 4D dynamic system (18)gives rise to a period-2 cycle of the outputs which is synchronized with the period2 cycle of the trade policies.

(A) Return map(B) Time trajectory

Figure 4. Birth of a period-2 cycle
We has assumed so far that there is no time lag in implementing the trade
policy. However, the government policy usually works with a time lag since there
is an inevitable delay between the subsidy decision and the actual payment whichis the result of the political process. To examine the eﬀect caused by the policy
delay, a 10-period lag is introduced such that the trade policy is switched atevery 10 periods. Simulation results are illustrated in Figure 5 where we use thesame parameter specifications with the only diﬀerence that the length of lag ischanged to 10 from zero. In Figures 5(A) and 5(B), the output dynamics shows
the cyclic behavior in the following way: it fluctuates around xCA for 10 periods
and then around xCB for the next 10 periods, after which it jumps back to theoriginal cyclic behavior. We have already seen that xCA and xCB are locallystable if the trade policy is fixed. The cyclic behavior around each stationary
point is a dumping oscillation. When the policy is changed in the middle of theconverging process, the trajectory changes its direction and starts approachinga new equilibrium. As a result, a new dumping oscillation is born, which isagain interrupted before arriving at the equilibrium by a change of the policy.
A n-period time lag of the trade policy creates a period-2n cycle. It fluctuatesaround one stationary point for n periods and then jumps to a neighborhoodof the other stationary point when the policy is changed. Then it fluctuatesaround the new stationary state for the next n periods and jumps back to theprevious neighborhood when the policy is changed again. This recursive processrepeats itself. Dynamics with time lag can be summarized as follows:
Theorem 6 If the firms are symmetric, the policies and the outputs are naively
15

adjusted and there exists a n-period policy lag, then the 4D dynamic system gives
rise to a period-2 cycle of the trade policy and a period-2n cycle of the output.

(A) Return map(B) Time trajectory
Figure 5. Birth of a period-20 cycle
If the policy is naively adjusted and the output is adaptively adjusted, thenessentially the same dynamics will be observed. In other words, as far as thepolicy is periodically changed, the output dynamics is aﬀected by these policy switching and exhibits periodic behavior no matter which formation of the
expectations is selected.
If the policy is adaptively adjusted, then the policy adjustment process converges to the stationary point according to Theorem 2. We have, however,qualitatively diﬀerent output dynamics, as will be seen shortly. In this case we
can suppose without loss of generality that the dynamic process of the trade
policy is rapid and the firms receive the stationary values of the trade policy
from the beginning of the output dynamic process. This assumption reduces
the 4D dynamic system to the 2D output dynamic system (16) with s1 = se1 ,s2 = se2 and se1 = −se2 :
⎧
r
y
⎪
0
⎪
x
=
− y,
⎪
⎪
c
+
se2
⎨
(19)
r
⎪
⎪
x
⎪
0
⎪
− x,
⎩ y =
c − se2
where c = c1 = c2 .6 The stability of system (19) depends on the actual cost
ratio,
c + se2(20)k=c − se26 It is possible to construct the dynamic system in terms of se . However, the results become
1qualitatively the same.

16

Given c, k increases from zero to infinity as se2 increases from −c to c. In thelinear model, local instability implies global instability. However this is not necessarily the case with nonlinear models because the nonlinearities may prevent
unstable trajectories from globally diverging. We restrict our analysis to theunstable case, henceforth, to examine what dynamics the nonlinear output
sys√
tem can generate. The stability of the output is violated if k ≤ 3 − 2 2 and theoutput trajectories are non-negative as far as k ≥ 4/25. Therefore, the output
stationary state becomes locally unstable but the trajectories are nonnegative
if
√
2
21
− c < se2 < −
c.
(21)
29
2
√
Here k = 3 − 2 2 if se2 is equal to this upper bound which is the threshold of theloss of stability and will be called the instability value and k = 4/25 if se2 is equal
to this lower bound, which is the threshold of loss of nonnegativity and will becalled the nonnegativity value. When se2 decreases from the instability value
to the nonnegativity value in the interval, the stationary state is destabilized
and goes to chaos through a period-doubling cascade as shown in Figure 6. We
summarize this result as follows:
Lemma 1. If the firms are symmetric and the trade policy is adaptively adjusted, then the naively adjusted output exhibits various dynamics ranging
from a period-2 cycle to chaotic fluctuations, depending on the value of
se2 in the interval bounded by the instability value and the nonnegativity
value.

Figure 6. One-parameter bifurcation diagram with respect to se2 .
Now we replace the naive expectation formation with the adaptive expectation formation by taking β 1 < 1 and β 2 < 1. The output dynamic system

17

is

¶
µr
⎧
y
⎪
0
⎪
x
=
(1
−
β
)x
+
β
−
y
,
⎪
1
1
⎪
c + se2
⎨
µr
⎪
⎪
⎪
⎪
⎩ y 0 = (1 − β 2 )y + β 2

¶
x
−
x
.
c − se2(22)

It is also shown in Puu (2003) that loss of stability under the adaptive expectation occurs when the cost ratio and the speeds of adjustment satisfy thefollowing relation:
1
1(k − 1)2
+
−1
(23)
=
4k
β1 β2
under which the determinant of the Jacobian matrix of the dynamic system(22) becomes unity. To examine what dynamics can be generated, we simulate
system (22) when the adjustment speeds are identical (β 1 = β 2 = β). Taking thesame parameter specifications as in Figure 6, Figure 7 shows the two-parameterbifurcation diagram in which the horizontal coordinate is the level of the optimal
subsidy to firm 2 (i.e., se2 ) and the vertical one is the identical adjustment speed(i.e., β). Solving (23) for β and substituting (20) yields the partition curve of
the parameter space (se2 , β),β=2(c − se2 )(c + se2 )
.
c2

(24)

Given c, for all (se1 , β) under the partition curve, the stability condition is satisfied and this stable region is colored in red in Figure 7. For all (se1 , β) above thecurve, the stability condition is violated but the nonlinearities of the dynamics
system prevent diverging trajectories. The diﬀerent colors of the regions correspond to the diﬀerent periods of periodic cycles up to period 16. The grayregions indicate that the period of the cycle is larger than 16 or chaos emerges.
This is summarize as:
Lemma 2. If the firms are symmetric and the trade policy is adaptively andrapidly adjusted, then the adaptively adjusted output exhibits various dynamics ranging from a period-2 cycle to chaotic fluctuations, dependingon the values of (se2 , β).

18Figure 7. Two-parameter bifurcation diagaram

4.2

Asymmetric firms: c1 6= c2

We now turn our attention to the asymmetric firms with c1 < c2 . We havealready seen that the dynamic process of the trade policy is stable and converges
to the optimal points given in (12),eL
se1 = −sL
2 + (c2 − c1 ) and s2 = s2 ,where 0 > sL
2 > c2 − 2c1 due to Assumption 2. The corresponding optimal
outputs are obtained by substituting these optimal subsidies into the expressionsin (15),c2 − sL(2c1 − c2 ) + sL
2
2
C
xC =
and
y
=
.
4c21
4c21
If we assume that the implementation of the trade policy has no time lags, thenthe output dynamic system with β 1 = β 2 = 1 is
⎧
r
y
⎪
0
⎪
x
− y,
=
⎪
L
⎪
c1 − (−s2 + c2 − c1 )
⎨
(25)
r
⎪
⎪
x
⎪
0
⎪
− x.
⎩ y =
c2 − sL
2

Substituting se1 and se2 into the actual cost ratio yieldsk=2c1 − c2 + sL2.
c2 − sL
2(26)

It follows that, given c1 and c2 , the actual cost ratio decreases to zero fromu(2c1 − c2 )/c2 > 0 if sL
2 decreases to its lower bound s2 = −(2c1 − c2 ) from19

zero. Depending on the value of sL
2 , the output dynamics can be destabilized.
Furthermore, due to the nonlinearities of (16), the output dynamics can exhibita rich dynamics if sL
2 is in the interval
c2 −

50
1
√ c1 .
c1 < sL
2 < c2 −
29
2− 2

(27)

L
The upper bound value√ and the lower bound value of s2 make the actual cost
ratio equal to 3 − 2 2 and 4/25, respectively.
When the symmetric adaptive expectation formation is adopted (i.e., β 1 =
β 2 = β < 1), the output dynamic system is
⎧
!
Ãr
⎪
y
⎪
⎪
−y ,
⎪ x0 = (1 − β)x + β⎪⎪
c1 − (−sL⎨
2 + c2 − c1 )
(28)
Ãr
!
⎪
⎪
⎪
x
⎪
0
⎪
−x .
⎪
⎩ y = (1 − β)y + β
c2 − sL
2

Solving (23) for β and substituting (26) yield the partition curveβ=L2(c2 − sL
2 )(2c1 − c2 + s2 )
2
c1

(29)

which divides the (sL
2 , β)-space into two parts: stable region in the right to thecurve and unstable region left.
It can be shown that dynamics generated by (19), respectively (22), are essentially the same as dynamics generated by (25), respectively (28). Introducing
the new variables C = c1 and S = c1 − c2 + sL
2 reduces (25) and (28) to (19) and(22), respectively. One system can be transformed to the other through variable changes. Thus both systems are topologically conjugate to each other andgenerate qualitatively the same dynamics. In particular, the instability value
and the nonnegativity value of (25) can be obtained from those values of (19)with C and S.
21
50
S = − C =⇒ sLc1
2 = c2 −
29
29
and
√
1
2
√ c1 .
S=−
C =⇒ sL
2 = c2 −2
2− 2
Furthermore the partition line, (24) with C and S, can be transformed to (29).
β=L2(c2 − sL
2(C − S)(C + S)
2 )(2c1 − c2 + s2 )=⇒ β =.
2
2
C
c1

The equivalence of the dynamic systems implies that (25) generates the same
dynamics as illustrated in Figure 6 with replacing the interval (21) with (27).
Similarly, the output dynamics by (28) is the same as illustrated in Figure 7 withreplacing the partition curve (24) with (29). We can summarize these results asfollows:
20

Lemma 3. If the firms are asymmetric, then the naively adjusted dynamic
system (25) starts the period-doubling bifurcation leading to chaos if sL
2decreases from the instability value of the interval (27) to the nonnegative value whereas the adaptively adjusted dynamic system (28) generates
complex dynamics involving chaos for (sL
2 , β) such asβ>L2(c2 − sL
2 )(2c1 − c2 + s2 ).
2c1

Lemmas 1 and 2 are concerned with the output dynamics of the symmetricfirms while Lemma 3 is concerned with the output dynamics of the asymmetricfirms. Notice that the results are essentially the same, hence, the production
cost diﬀerences do not aﬀect the asymptotic behavior of the unstable output
dynamics if the trade policy is adaptively adjusted. These results can be summarized as follows:
Theorem 7 If the optimal trade policy is asymptotically stable, then the output
dynamic system generates the same dynamics regardless of the symmetry orasymmetry of the firms.
The output dynamic system with adaptive expectation might lead to negative quantities in numerical simulations. To avoid such economically unfavorable phenomena, we repeated the simulations with the modified output dynamic
equations
∙ µr
¶¸
⎧
y
⎪
0
⎪
x
=
(1
−
β)x
+
βM
ax
0,
−
y
,
⎪
⎪
c1 + se2
⎨
(30)
∙ µr
¶¸
⎪
⎪
x
⎪
⎪
−x .
⎩ y 0 = (1 − β)y + βM ax 0,
c2 − se2

The results are shown in Figure 7. This is not the only way to prevent negative
quantities. Following Yousefi (2002), we can also use the alternative formulation
∙
µr
¶¸
⎧
y
0
⎪
⎪
x = M ax 0, (1 − β)x + β−y ,⎪⎪c1 + se2
⎨
(31)
∙
µr
¶¸
⎪
⎪
x
⎪ 0
⎪
−x .
⎩ y = M ax 0, (1 − β)y + β
c2 − se2

It is clear that the asymptotic behavior of (30) is diﬀerent from the asymptotic behavior of (31). Figure 8 shows the two-parameter bifurcation diagram
generated by (31). Apparently there are many diﬀerences between the bifurcation diagrams of Figures 7 and 8. However, our main finding that the adaptive
systems can generate rich dynamics still can hold in the bifurcation diagram

21

shown in Figure 8.

Figure 8. The two-parameter bifurcation diagram generated by (??)

5

Conclusion

In this paper, we construct a three-country model with two governments andtwo firms and consider the dynamic behavior of the sequential subsidy game inwhich the governments determine their optimal trade policies and then the firms
determine their optimal outputs. We first deal with the governments’ decision
process from static and dynamic points of view. We find that the cost diﬀerence
and the expectation formation of the governments are crucial in characterizing
the optimal trade policy. In short, if the firms are symmetric, then there areinfinitely many optimal policies (Theorem 1). A symmetric period-2 cycle ofthe trade policy emerges if naive adjustment process is adopted (Theorem 2)
and a trajectory converges to one of the optimal policies if adaptive adjustment
process is used (Theorem 3). If the firms are asymmetric, then a unique optimal
policy exits and is asymptotically stable regardless of the expectation formations(Theorem 4). We then deal with the output dynamics and demonstrate thatthe expectation formation of the government matters but the cost diﬀerence
does not matter. If the trade policy is adaptively adjusted, then an output
trajectory exhibits periodic cycle which is synchronized with the period-2 cycleof the optimal trade policy even if the Cournot output equilibrium is locallystable (Theorems 5 and 6). If the trade policy is adaptively adjusted, thencomplex output dynamics involving chaos emerges regardless of the expectation
formation (Theorem 7). Finally it is worth mentioning that complex dynamics
can be born under a small or even zero diﬀerence of the production costs in ourmodel while much larger diﬀerence is required to generate chaotic dynamics innonlinear duopoly models with isoelastic price function.

22

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Eaton, J., and G. Grossman, "Optimal Trade and Industry Policy under
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23