Controlling Cournot-Nash Chaos
Akio Matsumoto
Department of Economics
Chuo University742-1, Higashi-Nakano
Hachioji, Tokyo,192-0393
Japan
Tel: 81-(0)426-74-3351; Fax: 81-(0)426-74-3425
E-mail: akiom@tamacc.chuo-u.ac.jp
June 23, 2004

Abstract
The recently developing theory of non-linear dynamics shows that any economic model can generate complex dynamics involving chaos if its nonlinearities become strong enough. This study constructs a nonlinear Cournot
duopoly model, reveals conditions for generating chaos and then considers
controlling chaos. The main purpose of this paper is to demonstrate thatchaos generated in Cournot competition is in a double bind from the longrun perspective: a firm with a lower marginal production cost prefers a stable(i.e., controlled) market to a chaotic (i.e., uncontrolled) market while a firmwith a higher marginal cost prefers the chaotic market.

Acknowledgement
Financial supports from Chuo University (Joint Research Grant 0382)
and from Ministry of Education, Culture, Sports, Science and Technology(Grand-in-aid for Scientific Research (B)15330037) are highly appreciated.

1

Introduction

In his seminal paper, Day [1982] has introduced nonlinear dynamics into
economics and showed that even traditional economic dynamics models can
exhibit complex dynamics involving chaos when nonlinearities become strong
enough. Since then, a lot of eﬀorts have been devoted to investigate conditions for the existence of chaotic dynamics in economic dynamic models andto provide fruitful theoretical insights. Among others, it is demonstrated thatcompetitive markets can be chaotic, see Boldrin and Woodford [1990] andMajumdar et al. [2000]. This finding implies that the standard assumptionsof economic theory such as convex preferences and production technologies
are compatible with chaotic dynamics. In the recent literature, it is also
demonstrated that even oligopolistic markets may become chaotic; see Rand
[1978], Kopel [1995], Puu [2000] and studies included in Puu and Sushko[2002]. These works indicate that the higher-dimensional deterministic modelof a dynamic economy can explain various complex dynamic behavior of theeconomic variables. More recently, Matsumoto and Nonaka [2004] reveal
an interesting characteristics of economic chaos by showing that chaotic dynamics may be profitable from the long-run perspective in two-dimensional
output adjustment process of complementary goods.
In spite of these developments, unstable fluctuations have been regarded
as unfavorable phenomena in traditional economics that has mainly focusedon a stationary state. It is therefore a natural question of whether chaotic
behavior can be stabilized or controlled to make the state converging. In fact,
the research for such a direction has already begun and there already exist
several studies on controlling chaos, see Kopel [1997], Kass [1998], Bala et al.
[1998], Mendes and Mendes [2000], to name a few. In the existing literature,
however, not much has yet been revealed with respect to the economic implications of generating chaos and controlling chaos. In particular, it has notbeen determined which is preferable, controlling chaos or generating chaos.
The main purpose of this paper is to consider economic implications ofgenerating chaos as well as controlling chaos in the nonlinear duopoly model
developed by Puu [2000]. An equivocal characteristics of economic chaos will
be demonstrated. From the long-run point of view, one of the duopolists canbe beneficial and the other harmful in the chaotic market. Putting control
reverses the situation: the beneficial duopolist in the chaotic market becomes
disadvantageous and the harmful firm advantageous in the controlled market.
This implies that either way of generating chaos or controlling chaos is unable
to make both duopolists happy together.
The paper is organized as follows. Section 2 constructs a simple nonlinear duopoly model. Section 3 examines two statistical properties of chaotic
1duopoly map; the long-run average behavior and autocorrelation between
outputs. Section 4 considers controlling chaos. Section 5 provides summaryand concluding remarks.

2

Puu’s Model

Rand [1978] shows that a Cournot duopoly model with unimodal reaction
function can give rise to chaotic dynamics. Puu [2000] presents a possible economic underpinning which supports the unimodal reaction function. Since weconsider controlling chaos in Puu’s setting, we recapitulate the fundamental
structure of Puu’s model in this section.
The market demand is assumed to be isoelastic such that price p is reciprocal to the total demand Q,
1
p= .
(1)
Q
There are two firms, denoted by X and Y, producing the amounts of goods x
and y with constant marginal costs a and b, respectively. Goods are perfect
substitutes so that, provided demand equals supply, the total demand equals
the total supplies, Q = x + y. Their expected profits become accordingly,

x

Πx =
− ax,


x + ye(2) Πy = y − by,xe + ywhere ”e” denotes an expected value. The usual procedure to maximize profit
leads to unimodal reaction functions which construct the following dynamic
eprocess under the naive expectation formation (i.e., xet+1 = xt and yt+1
= yt ),
 xt+1 = f (yt ),(3)

yt+1 = g(xt ),where f (y) is the reaction function of firm X and g(x) is the reaction functionof firm Y. Both are specified asryt− yt ,f (yt ) =
a
(4)
r
xt
g(xt ) =− xt .
b2

1
As can be seen in Figure 1 below, f (y) has its maximum value 4aat1
1
y = 4a , and its domain should be restricted to the interval [0, a ] for nonnegative values of output, and so does g(x) with replacing a with b.1 When the
dynamic process is designed to map the maximum point to an interior pointof the interval, it can generate positive productions to both firms. Solving
1
1
g( 4a
) ≤ a1 and f ( 4b) ≤ 1b gives the upper and lower bounds of the marginal
cost b in terms of the marginal cost a,

425a ≤ b ≤ a.
25
4

(5)

If these inequalities are violated, the dynamic process (3) induces negative
output values.
As denoted as ”C” in Figure 1, the fixed point of this dynamic process,which we call the Cournot point, is the intersection of the reaction curves,
xc =

b
a
c
and
y
=
,
(a + b)2(a + b)2(6)

where superscript ”c” is attached to variables to indicate the one at theCournot point.2 Substituting these Cournot outputs in the profit functions(2) gives the profits earned at the Cournot point,
Πcx =

µ

b
a+b

¶2

and Πcy =

µ

a
a+b

¶2

.

(7)

Taking ratios of Cournot outputs (6) and Cournot profits (7) gives
xc
Πcx
b
and
=
=
yc
a
Πcy

µ ¶2
b
,
a

(8)

which we summarize in
Theorem 1 At the Cournot point, a firm with the lower marginal cost produces more output and makes more profit than a firm with the higher marginal
cost,
a T b implies xc S y c and Πcx S Πcy .
1

The ratio of the horizontal axis and the vertical axis in Figure 1 is appropriately
adjusted in order to emphasize the mound-shape of reaction functions.
2
The dynamic process has two fixed points: one is the trivial point, (0, 0), and theother is a non-trivial point, (xc , y c ). Our concern is on the nontrival point and thus nofurther consideration is given to the trival point.

3

We call a firm with the lower marginal cost an eﬃcient firm and one withthe higher marginal cost an ineﬃcient firm. Theorem 1 implies the natural
result that an eﬃcient firm dominates the market and is more profitable thanan ineﬃcient firm at the Cournot point. Firm X is eﬃcient and firm Y isineﬃcient in Figure 1(A) where xc > y c and Πcx > Πcy while the relativeeﬃciency is reversed in Figure 1(B) where xc < y c and Πcx < Πcy .

1ê4b

y

gHxL

yc

1êa

f HyL

y

f HyL
C

yc

C

gHxL

1ê4a

1ê4b

xc

x
1êb

xc

(A) a = 1, b = 4.

x1ê4a(B) a = 1, b = 0.2.

Figure 1. Reaction functions and Cournot points.
To examine the stability of the Cournot point, we make a linear approximation of the dynamic process at the Cournot point and construct the Jacobi
matrix,


b−a
 0
2a 
(9)
JC =  a − b,02bwhere the trace and determinant of JC are
trJC = 0 and det JC =(a − b)2.
4ab

It can be checked that the conditions (S2) and (S3) of Stability Result presented in Appendix A are satisfied. Thus loss of stability occurs when theabsolute value of a eigenvalue becomes unity, that is, when det JC = 1 or(a − b)2 = 4ab holds. The dynamic process, therefore, is stable if the marginal cost of firm Y falls inside the interval bounded by the two roots of thelast equation,√√(3 − 2 2)a < b < (3 + 2 2)a.
(10)
4

Since we are interested in controlling chaotic oscillations, we first derive thecondition under which the Cournot point is locally unstable. Notice that if
a = b, then both eigenvalues equal zero, so stability necessarily holds. So wemight assume that a 6= b. Taking into account the upper or lower bound for
economically feasible (that is, non-negative) production level given in (5) andthe stable interval defined in (10), we find that the unstable condition can be
expressed by the ratio of the marginal production costs and is split into twocases according to whether a is larger or smaller than b. Since one condition
is derived from the other (that is, two are reciprocal), we can make, withouta loss of generality, the following assumption:√
25bAssumption. b > a and (3 + 2 2) < ≤ .
a4Under Assumption, the Cournot point is unstable so that trajectories
starting from any points of a neighborhood of the Cournot point move away.
However the nonlinearity of the dynamic process prevents trajectories fromglobally diverging from the Cournot point. Trajectories are bounced backto the neighborhood soon or later but move away again. This process isrepeated again and again. That is, the dynamic process does not converge
to the Cournot point but keeps fluctuating within a limited region. Figure
2 depicts a bifurcation diagram with respect to the cost ratio under thecondition that the b/a > √1 as set in Assumption. It can be seen that as theratio increases from 3 + 2 2, the Cournot point becomes unstable and thenbifurcates to a stable period-2 cycle, generates the period doubling sequenceand leads to a two-piece chaotic attractor, which finally merges to a one-piece
attractor as the ratio approaches its upper bound, 25
.
4
xt

è!!!
3+2 2

25ê4

Figure 2. Bifurcation Diagram.
5

bêa

3

Statistical Properties

It is well-known that chaotic dynamics has two salient features: sensitivity
to the initial conditions and irregularity of trajectory. We examine thesefeatures of chaotic duopolies in this section. The former feature implies thateven a slightly diﬀerent choice of initial conditions can drastically alter thewhole future behavior of the trajectories. Therefore it is meaningless toinvestigate an individual chaotic trajectory. Instead, we examine the longrun average behavior of chaotic trajectories in the first half of this section.
The latter feature implies that it is diﬃcult to make precise expectations
about future values of variables along chaotic trajectories. Alternatively,
forecasting errors are inevitable in every period. A natural consequence isto reject the current predictor which shows systematic errors and then toalter the formation of expectations. Nevertheless such alternations are notadequately explored in Puu’s setting. We thus consider ”consistency” of thenaive expectations in the second half of this section.

3.1

Long-run Average Return

A consequence of the sensitivity is that even if two trajectories start fromsimilar initial conditions, they will be apart sooner or later and move in sucha diﬀerent and complicated way that it is diﬃcult to predict their long-run
behavior. One way to characterize such chaotic dynamics is to turn our attention to statistical or long-run behavior. Under rather weak mathematicalcondition, frequencies of chaotic trajectories may converge to a stable density function. Once the explicit form of the density function is constructed, it
can be shown that for any continuous function of a variable, its time averagealong the chaotic trajectory is equal to its space average.3 This implies that
it is possible to analytically calculate the long-run average behavior which is
independent from a choice of initial points. However, it is, in general, diﬃcult
to construct such an explicit form of the density function. We thus numerically calculate the long-run average behavior over suﬃciently long period of3

Let xt+1 = θ(xt ) be a dynamical process where xt is a variable at time t. Suppose
the frequencies of the trajectory {xt }∞t=0 converges to a density function ϕ. Then for any
continuous function f and for any initial point x0 ,
T −1

1 X
lim
f (θt (x0 )) =T −>∞ T
t=0

Z

f (x)ϕ(x)dx,

where θt = θt−1 · θ and θ0 = 1. See, for example, Chapter 8 of Day [1994] for moredetails.

6

time as a proxy for the analytical value of the long-run average behavior.
Given the dynamic process (3) with Assumption, the average profit oftime-serise with T periods is defined,1XΠ̄i =Πi (xt , yt ), i = x, y.
T t=0
T −1

Figure 3 illustrates the results of the numerical simulations of the long-run
average profits of firm X (left) and firm Y (right) (denoted as π̄x and π̄ y )against variations of the marginal cost ratio. For comparison, graphs of theCournot profits are also depicted there. In each illustration, the ratio isincreased in steps of 0.002 from 5.75 to 6.25, and for each of these ratio
values, the average is calculated from the last 1, 000 out of 5, 000 iterations.
It can first be observed that the average profit for each√ firm is identical to
its Cournot profit when the ratio is less than 3 + 2 2(' 5.8) (that is, theCournot point is stable). It is also clear that the average profit is less thanthe Cournot profit for firm X (i.e., the eﬃcient firm), and greater√ for firm
Y (i.e., the ineﬃcient firm) when the ratio is greater than 3 + 2 2 but lessthan 6.25. The numerical simulations indicate the following results:
Theorem 2 When the Cournot point is unstable, the long-run average profitof the eﬃcient firm is less than the Cournot profit while the long-run average
profit of the ineﬃcient firm is more than the Cournot profit,√b≥ 3 + 2 2 implies Π̄x ≤ Πcx and Π̄y ≥ Πcy .
aTheorem 1 and Theorem 2 imply that eﬃcient firm X dominates themarket but is harmful in the unstable market in the sense that its long-run
average profit is less than its Cournot profit. On the other hand, ineﬃcient
firm Y has only a small share of the market but is beneficial in the sense thatits long-run average profit is higher than its Cournot profit.

7

Profit

Profit

p xC

êê
py
êê
px

è!!!
3+2 2

6.25

b êa

è!!!
3+2 2

p yc
6.25

bêa

Figure 3. Long-run average profits and Cournot profits.

3.2

AutoCorrelations

We apply a formation of naive expectations to build dynamic process (3).
e
The formation is given by xet+1 = xt and yt+1
= yt , that is, today’s expectedvalue is equal to yesterday’s realized value. It is the simplest formation of
more general backward-looking expectations where the expected price is afunction of the past prices. A conceptual problem with native expectations
is that it may lead to systematic forecasting errors, that is, firms are making
wrong expectations along chaotic trajectories. On the other hand, it has beenshown that irregularity of chaotic trajectories is indistinguishable from randomness of a stochastic process if autocorrelations between outputs are zeroat all lags (see, for example, Sakai and Tokumaru [1980]). In this case, there
is no reason to switch to another formation of expectations. In the simple
setting of a cobweb economy, Hommes [1996] explores whether a chaotic cobweb model with the backward-looking expectations can have such statisticalproperty, and numerically shows that under the naive expectations, prices
are correlated with past prices if the supply function is monotonic and areuncorrelated if the supply function is non-monotonic (e.g., piecewise linear).
On the contrary, we show that the autocorrelation coeﬃcients of the naive
expectations are significantly diﬀerent from zero even if the dynamic process
is nonlinear enough to produce chaotic fluctuations
Following Hommes [1996], we define consistency of expectations by meansof the autocorrelation function (ACF) of expectation errors. Expectation
errors are given by
et = xt − xet .
8The (empirical) autocorrelation coeﬃcients ρk of expectation errors are defined asckρk = , −1 ≤ ρk ≤ 1,
c0
with
1 PN
et ,
ē = limN→∞
N t=1

1 PN
(et − ē)(et+k − ē), k ≥ 0.
N t=1
An expectation formation is called consistent if the autocorrelation coeﬃcients of the expectation errors are zero for all k ≥ 1; weakly consistent if
there exists a K ≥ 2, such that the autocorrelation coeﬃcients of the expectation errors are zero for all k ≥ K; inconsistent if it is not weakly consistent.
Consistent expectations mean that firms are not able to distinguish chaotic
fluctuations from random fluctuations so that they have no reason to change
their beliefs. On the other hand, inconsistent expectations mean that they
can make distinction and therefore change expectation formations accordingly.
The dynamic process with the naive expectations (3), which shows periodic cycles and irregular behavior as seen in Figure 2, consists of two
independent iterative process, f (yt ) and g(xt ). Iterations produced by thetwo-dimensional map are essentially the same as the ones by the combined
one-dimensional map derived by substituting one into the other. Only diﬀerence is that one period of the combined map is equivalent to two periods interms of the originally introduced lag. We investigate forecast errors generated by the combined one-dimensional map f (g(x)) with the initial conditionx0 = 0.1 and parameters a = 0.1 and b = 0.625. Figure 4 gives the simulation
results. Chaotic time series of output x is depicted in the left and its sample
autocorrelation coeﬃcients at the first 20 lags in the right. A period-doublingbifurcation diagram depicted in Figure 2 implies that output trajectories converge to a 2-piece chaotic attractor. The chaotic time series jumps back andforth between two intervals. Accordingly, the corresponding ACF of expectation errors has ”period” 2, with strong negative autocorrelations at odd lags
and strong positive autocorrelations at even lags. Therefore the numericalsimulations indicate the following:ck = limN→∞

Theorem 3 Naive expectations are inconsistent so that there exists systematic forecasting errors in the dynamics process (3).

9

x

ACF
1

1.5
1.25

0.5

1
0.75

5

0.5

10

15

20

lag

-0.5

0.25
20

40

60

80

100

-1

t

(A) Chaotic Trajectory(B) Auto Correlation

Figure 4. Output fluctuations and corresponding expectation errors.

4

Controlling Chaos

In this section, we apply two distinctive control methods, the adaptive control
method advocated by Huang [2002] and a generalized version of the poleplacement method developed by Ueta and Kawakami [1995], for the output
adjustment process to stabilize the Cournot point embedded in the chaotic
attractor. Both methods are known for a long time in applied mathematics
The adaptive control method, as called the Mann’s iteration method, hasbeen investigated very intensively in the numerical analysis literature. The
pole-placment method is also a definite stabilizing approach in system design,
and it has many important applications such as constructing observers.

4.1

Adaptive Control Method

Naive expectations are inconsistent as it has been shown in the last section.
Sooner or later, each of duopolists might discover significant autocorrelationcoeﬃcients by observing time series data and may change his beliefs on the
opponent behavior. In this subsection, we suppose that duopolists change
their beliefs to adaptive expectations from naive expectations. Optimal output in the next period is determined as a weighted average of current period
expectation of the rival firm and current period’s actual output. The dynamic
process (3) turns to be xt+1 = (1 − λx )xt + λx f (yt ),(11)
yt+1 = λy g(xt ) + (1 − λy )yt ,10

where λx and λy are adjustment parameters of firms X and Y. Since themodified (adaptive) process (11) is back to the original (naive) process (3)for λx = λy = 1 and gives rise to no dynamics for λx = λy = 0, the adjustment
parameters are assumed to take intermediate values between zero and unity.
It is well-known that adaptive expectations are an another example ofbackward-looking expectations and useful to stabilize a unstable economic
process. It can be shown that adaptive expectations by the firms can be described by a four-dimensional system, where the eigenvalues of the Jacobian
are zero with multiplicity 2, and the eigenvalue of the Jacobian of (11). In thissense the two processes can be considered equivalent to each other as shownin Appendix B. Puu [2000] also extends his naive expectations model to theadaptive expectation model. However, he draws his attentions to a possibility that the adaptive process can have fractal attractors in two dimensional
embedding parameter space. On the other hand, Huang [2001] generalizes
the concept of adaptive expectations to adaptive adjustment mechanism insuch a way that economic variables are directly adjusted adaptively to equilibrium point. Following his formulation, we consider an adaptive feedback
mechanism as a means of controlling the unstable chaotic process.
The goal of implementation of the adaptive adjustment is to stabilize thechaotic dynamic process directly through the adjustment parameter of each
duopolist. We retain Assumption so that the Cournot point is unstable in
the naive process (3). It can be checked that the adaptive process (11) has
exactly the same fixed points as the naive process (3). In the following, we areinterested in stabilizing the unstable dynamic process by using perturbationsof control parameters.
The Jacobi matrix of the adaptive process (11) evaluated at the Cournot
point is
λx (b − a)
 1 − λx2aJA =  λ (a − b).
y1 − λy2bwhere a trace and determinant of JA are
trJA = 2 − (λx + λy ),detJA = (1 − λx )(1 − λy ) +λx λy (a − b)2.
4ab

According to Stability Result in Appendix A, the domain of stability of11equilibria is determined by the following three conditions,

4cλx


detJA < 1,
⇒ λy >

(1 + c)2 λx − 4c






(1 + c)2
λx λy > 0,
>
trJ
−
1
⇒
detJ
A
A

4c






2


 detJA > −trJA − 1 ⇒ 2(2 − λx − λy ) + (1 + c) λx λy > 0,
4c

(12)

where c = ab is the marginal production cost ratio. Since we assume thatthe marginal costs are positive, and the adjustment parameters are between
zero and unity, the second and third inequalities of (12) are always true.
That is, there are no situations in which iterative dynamics either diverges
or generates the period-doubilng bifurcation. Thus the domain of stability of
the Cournot point is defined only by the flutter boundary, det J A = 1, anddepicted as a gray-coloured region in Figure 5. It is divided into three parts,
a light-gray region and two dark-gray regions by the steeper real line (i.e.,
λy = cλx ) and flatter real line (i.e., i.e., λy = 1c λx ) on which the discriminantof the characteristic equation is zero. Eigenvalues of JA are real for parameters in the dark-gray regions and complex for the light-gray region. Thus a
trajectory monotonically (oscillatory) converges to the Cournot point when apair of adjustment coeﬃcients are in the dark (light)-gray region. The naive
process, which is unstable under Assumption, can be stabilized by appropriate choice of the adjustment coeﬃcients.
ly

1
0.8
0.6
0.4
0.2

0.2

0.4

0.6

0.8

1

lx

Figure 5. Domain of Stability with c = 25.
4
12

The adaptive adjustment cannot fully control chaotic trajectories when
the adaptive coeﬃcient are close to 1. In fact, the white region located at thenorth-east part of Figure 5 is the unstable region which is separated from thestable region by the detJA = 1√locus. The locus passes through the north-east
corner (1, 1) when c = 3 + 2 2 for which√ the loss of stability occurs, shifts
upward from the corner when c < 3 + 2 2 (alternatively,
the unstable region√
disappears) and shifts downward when c > 3 + 2 2 (the unstable region
becomes larger). Since c = 6.25 is the maximum value for the original process
under economic feasibility, the white region in Figure 5 reaches the maximum
size. Although it is only a small part of the whole region, its existenceindicates incapability or limitation of the adaptive adjustment mechanism.
We performed two numerical simulations to see how adaptive adjustment
controls the unstable Cournot point. Parameter values were fixed as a =1 and b = 6.25 (i.e., c = 6.25) henceforth. We then considered diﬀerent valuesof the adjustment coeﬃcients. It was shown that a chaotic trajectory could
be stabilized in the first simulation and could not in the second simulation.
In the following figures, ”red” means a trajectory generated by the original(naive) process and ”blue” a trajectory by the controlled (adaptive) process.
We selected λx = λy = 0.8 in the first simulation. In the left part of
Figure 6 in which return maps are depicted, we observe that two trajectories
start at the same initial point, x0 = 0.01 and y0 = g(x0 ), and the blue
trajectory converges oscillatory to the Cournot point while the red trajectory
keeps fluctuating. In the right part of Figure 6 in which the correspondingtime series are presented, we see that the blue trajectory almost reaches the
Cournot point a little bit more than 20 periods of time.
yt

yt

yc

yc

xc

xt

5

10

15

20

t

Figure 6. Example of controlled trajectory.
We increased the value of the adjustment parameters to λx = λy = 0.96 in
the second simulation. In Figure 7 in which two return maps are illustrated,13

we can see that both trajectories start at the same initial point as in the firstsimulation and the red trajectory on the right part of the figure keeps fluctuating while the blue trajectory does not converge to the Cournot point either
but approaches a quasi-periodic cycle. This is an example of uncontrollable
chaotic behavior.
yt

yt

yc

yc

xc

xt

xc

xt

Figure 7. Example of uncontrollable trajectory.
The adaptive control can be considered as a ”private” or ”individual”
method to control a chaotic market. Each firm has a wide spectrum of thecontrol power. A larger adjustment coeﬃcient means to place an emphasis on
the current profit maximizing output in order to determine the output in thenext period, while a smaller one on the current realized output. Dependingon the degree of emphasis, firms can gradually approach the Cournot pointin either a monotonic or oscillatory way. As shown in the previous section,
firms are able to learn by observing actual output time series that they makeforecasting errors and thus have incentives to change their expectation formations. The shape of the stability domain illustrated in Figure 5 implies
that the chaotic market can become controllable through individual eﬀorts
by changing naive expectations to adaptive expectations. As it was also
shown in the previous section, the eﬃcient firm is harmful and the ineﬃcient
firm is beneficial in the chaotic market. Since it prefers the chaotic market,
the ineﬃcient firm possibly continues to form naive expectations (that is,
λy = 1). Even so, the eﬃcient firm alone can stabilize the chaotic market if
it selects the adjustment coeﬃcient suﬃciently small. In particular, the market is stabilized for any pair of the adjustment coeﬃcients such that λy = 14cand λx < (1−c)
2 for which the upper horizontal line crosses the detJ = 1
locus shown in Figure 5.

14

4.2

Pole Placement Method

In this subsection, we apply the Ueta-Kawakami method for fully stabilizing
the unstable Cournot point. It is a generalized version of the RGOD method,4a conventional state feedback method to control chaos. A general idea is toput control when a trajectory wandering in the chaotic attractor gets close
to a Cournot point. In particular, we stabilize the Cournot point by using
perturbations of parameters as the control input.
In order to apply linear control theory, we linearize the dynamic process(3) in a neighborhood of the Cournot point with respect to its variables andparameters:
¶
µ
¶
µ
¶
µ
∆xt+1
∆xt
∆a
,
(13)
=A
+B
∆b
∆yt+1
∆yt

where ∆ indicates a diﬀerence of a variable or parameter from the corresponding stationary value. In (13), A is the Jacobi matrix, which is identicalto JC introduced in Section 2,


b−a
 0
2a 
(14)
A= a−b,02band B = (Ba Bb ) shows the eﬀect of the variations of parameters a and b on
the state variables, which is given by




1
0
−
.
1
Ba =  2a(a + b)  and Bb = (15)−02b(a + b)

For stabilizing the unstable Cournot point, we construct the state feedback
¶
µ
¶
µ
∆a
∆xt
,
(16)
=C
∆yt
∆b

where C = (cij ) (i = a, b, j = 1, 2) is a constant matrix to be determinedlater. Substituting (16) into (13) yields the following homogeneous system
µ
¶
µ
¶
∆xt+1
∆xt
= JP
with JP = A + BC.
(17)
∆yt+1
∆yt
Consequently, the corresponding characteristic equation is|JP − µI| = 0
4(18)

RGOD stands for Romeiras, Grebogi, Ott and Dayawansa. See their paper, Romeiras
et al. [1992].

15

It is now clear that the system can be stabilized if a suitable gain matrix
C can be found such that the Jacobi matrix, JP , has eigenvalues inside theunit circle. Thus the stabilization problem of an unstable Cournot point ofsystem (3) is reduced to the well-known pole assignment problem of linear
control theory.
We first check whether the system is controllable (i.e., whether such C
can be found), otherwise choosing control gain is meaningless. The eigenvalue placement theorem of linear control theory implies that if, in our case,rank(Ba | ABa ) = 2 or rank(Bb | ABb ) = 2, then (3) is stabilizable.5 It
follows from (14) and (15) that


1
−
0
 2a(a + b)
rank (Ba | ABa ) = rank 

a−b
0
−
4ab(a + b)
= 2,
and




rank (Bb | ABb ) = rank 
= 2.

0
−

1
2b(a + b)a−b
4ab(a + b) 
0Therefore the system is controllable for either parameter, a or b.
If we consider a as a control parameter,6 we can construct the state feedback,
¶
µ
¡
¢ ∆xt
∆a = ca1 ca2
.
(19)
∆yt
The characteristic equation of controlled system is
¯
¯
¯
ca1
(b2 − a2 ) − ca2 ¯
¯ −
¯
−µ
¯
¯
2a(a + b)| A + Ba C − µI | = ¯ 2a(a + b)
¯
a−b
¯
¯
¯
¯
−µ
2b
= µ2 − trJP µ + detJP = 0,

where
trJP = −56ca1(a − b)(b2 − a2 − ca2 )and detJP =.
2a(a + b)
4ab(a + b)

See Theorem 1 of Ueta and Kawakami [1995].
Choosing b as the control parameter generates qualitatively the same result.

16

If trJP and detJP are in the shaded triangular region shown in Figure A
of the Appendix, the module of the eigenvalues is less than unity and thus
the controlled system becomes stable. The control cai can be solved by thecharacteristic equation. In the special case of trJa = detJa = 0 (i.e., thedean beat control), control parameters are given by
ca1 = 0 and ca2 = b2 − a2 .
Thus the controlled system is

r
yt


x
− yt ,=t+1
a + v(yt )

r


xt


 yt+1 =− xt ,bwhere, given arbitrary small value of ², 2
 (b − a2 )(yt − y c ) if | yt − y c |≤ εv(yt ) =0otherwise

(20)

(21)

(22)

The small value of ² means that the application of linear control theory
succeeds only in a suﬃciently small neighborhood around the Cournot point.
According to (22), the controlled system turns on only if values of (xt , yt )are inside the ²-neighborhood of the Cournot point and the original system
turns on otherwise.
We also present two numerical simulations in which the unstable Cournot
point is stabilized through the feedback control. Setting ² = 0.01, we choose
diﬀerent initial points on the reaction curve of firm Y ; x0 = xc − 0.095 andy0 = g(x0 ) in the first simulation and x0 = xc − 0.011 and y0 = g(x0 ) in
the second simulation. As in the previous cases, the red trajectory is theuncontrolled one and the blue is the controlled one. Figure 8 illustrates thereturn maps and time series of the first simulation. There we superimpose
the blue trajectory on the red trajectory. In the right part of the figure,
we can see only the blue trajectory in the first 6 periods, which means that
two trajectories start from the same initial point and take exactly the samevalues in the first 6 periods. In the left part it can be seen that the blue
trajectory is inside the ²-neighborhood (that is, in the ellipse circle) in the
6th period. The dynamic system is now switched to the controlled one (21)from the uncontrolled one (3). The eﬀect of control is clearly seen in theright part of Figure 8. The controlled blue trajectory jumps to the Cournot
17point in the next period and stays there since then while the uncontrolled
red trajectory moves oscillating away from the Cournot point. Figure 9
presents uncontrolled and controlled trajectories of the second simulation.
Since the blue trajectory is seen to converge to the Cournot point, the control
also works eﬀectively in the second simulation. However, on the contrary
to the first simulation, it takes longer periods of time until convergence isaccomplished. In fact, it takes more than 70 periods for the controlled system
to be turned on. These simulations show that although almost all initialconditions lead to controllable trajectories, the time until control is achieved
has initial-point dependency.
yt

yt

yc

yc

xc

xt

5

10

15

20

25

40

60

80 100t

Figure 8. Rapid Control.
yt

yt

yc

yc

20

40

60

80 100t

20t

Figure 9. Slow Control.
The feedback control may be thought as a ”public” method in the formof government interventions to stabilize a chaotic market. As defined in (21),18the control directly aﬀects the marginal cost of firm X. The form of intervention is, however, critically dependent on the behavior of firm Y. Sinceb > a due to Assumption, the sign of v(yt ) can be either positive or negative
according to yt is larger or smaller than y c . The government, therefore, introduces a production tax aiming at decreases in firm X 0 s production if firm
Y over-produces its output (i.e., v(y) > 0) and gives a production subsidy
aiming at increases firm X 0 s production if firm Y under-produces its output(i.e., v(y) < 0). Admittedly, the feedback control in terms of economic policy
is restrictive in the sense that the government is required to possess enormous
amount of information on not only firms’ production level but also on theevolution of the economy to make an accurate decision. Further, we naturally
raise the policy question of whether government assistance should be targeted
toward a weaker firm or a stronger firm. In the numerical example whereb > a is assumed, firm X is thought to be stronger because it dominates themarket share and earns the larger profits. It may seem paradoxical that the
government is more favour for the stronger firm and stabilizes the chaotic
market at the expense of profits of the weaker firm. However, the example
does help us to focus on an issue which has not been adequately explored:
if the chaotic market is controllable, which firms should be more favoured.
Finally we mention that the controllability of the production level in N-firmoligopolies linear price function has been early examined in Okuguchi and
Szidarovsky [1999]

5

Summary and Concluding Remarks

Our purpose of this study is to consider economic implications of generating
as well as controlling chaotic fluctuations in economic models. To this end,
we use the nonlinear duopoly model developed by Puu [2000] and carry out
the following results.
(I) A firm with a lower marginal cost of production (i.e., eﬃcient firm)
dominates the market and yields more profits than a firm with a higher
marginal cost (i.e., ineﬃcient firm) at the Cournot point.
(II) In case of b > a, economically feasible periodic as well as aperiodic
fluctuations involving chaos√ are evident when the ratio of marginal.
costs are higher than 3 + 2 2 but less than or equal to 25
4(III) From the long-run point of view, the eﬃcient firm is harmful and theineﬃcient firm is beneficial in the chaotic market because the long-run

19

average profit of the former is less than its Cournot profit, and the oneof the latter is larger.
(IV) The naive expectation formation is inconsistent in the sense that the
autocorrelations between outputs are non-zero so that firms have an
incentive to change their expectation formations.
(V) When firms adaptively forecast their rival’s behavior, the chaotic market
can be stabilized unless their adjustment parameters are close to unity.
(VI) Economic policy in the form of government interventions (subsidy or
tax on production) can fully stabilize the chaotic market.
(I) and (III) imply that the eﬃcient firm prefers the stable market while
the ineﬃcient firm prefers the chaotic market. (V) and (VI) imply that the
chaotic market can be stabilized by either individual eﬀort or public eﬀort.
Chaos in Cournot competition is in a double bind. In other words, generating
chaos is favoured for the ineﬃcient firm and not for the ineﬃcient firm;
controlling chaos is favoured for the eﬃcient firm and not for the ineﬃcientfirm. In short, a loser now in one market will be later to win in the othermarket.

Appendix A
In this appendix, we derive stability results on a two-dimensional discretemap, xt+1 = φ(xt , yt ),(A-1)

yt+1 = ϕ(xt , yt ),where xt and yt represent the components of the iteration process at time t.
We assume that the map has an equilibrium point, x∗ and y ∗ . To investigate
its stability, we make a linear approximation in the neighborhood of theequilibrium point,
µ
¶
µ
¶
∆xt+1
∆xt
=J
(A-2)
∆yt+1
∆yt
where ∆xt = xt − x∗ and ∆yt = yt − y ∗ denote diﬀerences of variables attime t from the equilibrium point, and J is the Jacobi matrix evaluated at20

the equilibrium point,∂φ(x∗ , y ∗ )∂xJ =  ∂ϕ(x
∗ ∗
,y )
∂x


∂φ(x∗ , y ∗ )∂y∂ϕ(x∗ , y ∗ )  .
∂y

(A-3)

The eigenvalues of the Jacobi matrix are the solutions of the characteristic
equation,
λ2 − trJλ + detJ = 0
(A-4)
where
trJ =

detJ =∂φ(x∗ , y ∗ ) ∂ϕ(x∗ , y ∗ )
+
,
∂x
∂y

∂φ(x∗ , y ∗ ) ∂ϕ(x∗ , y ∗ ) ∂ϕ(x∗ , y ∗ ) ∂φ(x∗ , y ∗ )
−
.
∂x
∂y
∂x
∂y

It has been demonstrated that the equilibrium (x∗ , y ∗ ) is asymptotically stable if both eigenvalues have modulus smaller than one,
|λ1 | < 1 and |λ2 | < 1.

(A-5)

Due to the outcome of the general Routh-Hurwitz stability condition, we
have the following result which ensures these inequality conditions. Since
these results are straightforward, we still omit a proof just although we often
use these in the analysis.
Stability Result: The polynomial λ2 − trJλ + det J has roots less thanunity in absolute value if and only if(S1) detJ < 1,(S2) detJ > trJ − 1(S3) detJ > −trJ − 1.
Stability Result is graphically summarized in Figure A in which theshaded triangular region shows the stable region satisfying the conditions(S1), (S2) and (S3). If trJ and detJ of (A-4) are placed in the dark-grayregion, the eigenvalues are real and thus the equilibrium is monotonicallystable. If they are in the light-gray region, the eigenvalues are complex
and thus the equilibrium is oscillatory stable. Two regions are separated2
by detJ = (trJ), the locus of trJ and detJ for which the discriminant is2
zero. Boundaries of the stable region are constructed from the following
21

three lines: the flutter boundary with the equation detJ = 1, the divergenceboundary with the equation detJ = trJ − 1, the flip boundary with theequation detJ = −trJ − 1.
On the divergence boundary at least one of the eigenvalues is equal to 1.
Crossing this boundary allows trajectories to approach infinity. On the flip
boundary at least one of the eigenvalues is equal to −1. Each point on theflip boundary corresponds to a two-periodic cycle, and movement outside thedomain of stability generates the Feigenbaum type period doubling sequence,leading to chaos. On the flutter boundary, |λ1 | = 1 and |λ2 | = 1. It is possible
to describe the types of bifurcation at all points of the flutter boundary.
The equality condition |λ1 | = |λ2 | = 1 means that λ1,2 = e±i2πθ with some0 ≤ θ ≤ 1. Therefore,
trJ = λ1 + λ2 = 2 cos 2πθ.
The dynamic system can generate a periodic trajectory on the flutter boundary according to the value of trJ. In particular, it generates a period-3 cycle
when trJ= −1, a period-4 cycle when trJ = 0, a period-5 cycle when√5−1
trJ = 2 , and so on.

detJ

1

-2

2

trJ

-1

Figure A Stable Region.

Appendix B
In this appendix, we show the process with adaptive adjustment is equivalent to the process with adaptive expectations.
22

The dynamic process with adaptive expectations is four-dimensional andgiven by
xt+1 = f (λy yt + (1 − λy )yte )
yt+1 = g(λx xt + (1 − λx )xet )(B-1)
xet+1 = λx xt + (1 − λx )xet
 e
yt+1 = λy yt + (1 − λy )yte

Its Jacobian is
0 f 0 λy0f 0 (1 − λy )
 g 0 λx0 g 0 (1 − λx )
0
.
J =
 λx

0
1 − λx00λy01 − λy(B-2)

Let the eigenvector be denoted by (x, y, u, v). Then the eigenvector equationsaref 0 λy y + f 0 (1 − λy )v = µx,g 0 λx x + g 0 (1 − λx )u = µy,(B-3)λx x + (1 − λx )u = µu,
λx y + (1 − λy )v = µv.

Subtracting the third equation multiplied by g0 from the second equation andalso subtracting the fourth equation multiplied by f 0 from the first equationyieldµy = µug 0 and µx = µvf 0 .
Dividing both equations by µ where µ 6= 0 is assumed gives
y = ug 0 and x = vf 0
which are substituted into the third and fourth equation of (B-3) to obtain
λx vf 0 + (1 − λx )u = µu,

(B-4)

0

λy ug + (1 − λy )v = µv.
This is equivalent to eigenvalue problem of the following matrixµ¶1 − λxvf 0.
ug 0
1 − λy(B-5)(B-5) is the same as the Jacobian of (11). Therefore the dynamic processwith adaptive adjustment is equivalent to the dynamic process with adaptive
expectations.

23

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