Discussion Paper Series No.51
Complexity yields benefits:
An example of a heterogeneous duopoly market
Yasuo Nonaka
October 27，2003

Complexity leads benefits :An example of a heterogeneous duopoly market
Yasuo Nonaka
Graduate School of Economics, Chuo University742-1, Higashi-Nakano,Hachiouji,
Tokyo, 192-0393, JAPANE-mail: q9111007@grad.tamacc.chuo-u.ac.jp
Abstract
This paper investigates an economic implication of chaotic ﬂuctuations. To
this end, it constructs a nonlinear, discrete-time Cournot model in which duopoly
ﬁrms are heterogeneous and have U-shaped reaction functions with diﬀerent microeconomic foundations. Due to the U-shaped reaction functions, output adjustment process generates chaotic ﬂuctuations. This paper demonstrates that
duopoly ﬁrms can eventually beneﬁt from chaotic output adjustment more thanfrom all possible equilibria when external eﬀects on production are asymmetric. The result indicates a possibility that chaotic ﬂuctuations are beneﬁcial forheterogeneous markets.

1

Introduction

Research in the past two decades has demonstrated that a Cournot adjustment processof output may be chaotic if reaction functions are non-monotonic; see Rand [14], Danaand Montrucchio [4] and Witteloostuijn and Lier [15] (W-L hereafter), to name a few.
At least two questions arise naturally: (i) Under what economic circumstances dooligopoly ﬁrms react non-monotonically to a change in the rival’s behavior? (ii) Is
chaotic adjustment of output a favorable economic phenomenon?
We have already had two possible answers to the ﬁrst question. One is given by
Puu [12, 13] who shows that proﬁt maximization with linear production technologies
and a hyperbolic market demand may result in unimodal reaction functions. The otheris by Kopel [8] who conﬁrms that, under a linear demand function and nonlinear costfunctions, proﬁt maximization also generates unimodal reaction functions. The second
question is partly answered for a one-dimensional dynamic process. Huang [7] uses
a cautious cobweb model and gives an aﬃrmative answer that perpetual ﬂuctuations
may be preferable to a competitive equilibrium from the long-run perspective. Kopel[8], on the other hand, gives a negative answer. He constructs a simple model of evolutionary dynamics to show that various performance measures such as aggregate proﬁts,
1

aggregate sales revenues, mean sales and the rate of return indicate the inferiority ofchaotic dynamics to equilibrium on the average. Matsumoto [9] provides an equivocal
result in a simple exchange model with two agents and two goods that the long-run
average performance measure (i.e., utility or proﬁt) taken along a chaotic trajectory
can be greater than the corresponding measure calculated at an equilibrium for one
agent but less for the other.
In this paper, we aim to provide an answer to the second question in a multidimensional dynamic process. In particular, we consider the long-run statistical behavior of Cournot duopoly ﬁrms. This paper is a continuation of Matsumoto and Nonaka [10] (M-N hereafter) which also investigates an economic implication of chaotic
adjustment of Cournot duopoly. M-N demonstrate through an example that chaotic
dynamics can be more beneﬁcial than stationary states in terms of the long-run average
proﬁts. They provide a weak result in which chaos is beneﬁcial from the viewpoint ofindustry proﬁts. In other words, the sum of the long-run average proﬁts of duopoly
ﬁrms taken along chaotic trajectories is larger than the sum of proﬁts obtained atstationary states. In this paper, however, we will provide the stronger result that foreach of duopoly ﬁrm, the long-run average proﬁt is strictly larger than any possible
equilibrium proﬁts if ﬁrms are heterogeneous and asymmetric.
The paper is organized as follows. Section 2 constructs a dynamic model of Cournot
duopoly. Section 3 examines existence and stability of equilibria numerically as wellas analytically. Through some numerical experiments, Section 4 investigates the longrun outcome of the output adjustment of duopoly ﬁrms. Section 5 gives concluding
remarks.

2

ModelConsider a market in which two ﬁrms, ﬁrm 1 and ﬁrm 2, compete by producing homogeneous outputs x and y. Assuming that inverse demand function is linear anddecreasing:p = a − b(x + y),

a > 0 and b > 0.

(1)

Without loss of generality, we normalize the set of outputs of ﬁrms to a unit intervalI ≡ [0, 1]. Equation (1) implies that the demand of one ﬁrm is externally aﬀectedby the other’s output level and this external eﬀect is always negative. In general,a downsloping demand curve implies a negative external eﬀect (negative externality
hereafter).
Our primary interest is in the cases where an externality exists not only on demand but also on production. To describe production externality we follow Kopel’s
approximation [8] and make the following assumption.
Assumption 1 Each firm has a marginal cost of production that is constant withrespect to its own output but varies with respect to the rival’s output. Let Ci be the costfunction of firm i. ThenC1 (x, y) = c1 (y)x and C2 (x, y) = c2 (x)y2(2)

where ci denotes the marginal cost of firm i.
We specify each ci as follows:c1 (y) = a − by − 2b(αy − α + 1)2
c2 (x) = a − bx − 2b(βx − 1)2 .

(3)The last terms in (3) imply that duopoly ﬁrms have marginal costs of production
which are unequal nonlinear functions of rival’s output in diﬀerent manners. Therefore
production externalities are nonlinear for both ﬁrms and heterogeneous between theﬁrms. In particular, if the parameters have relatively high values, both of the marginal
costs are unimodal to the rival’s output. That is, for each ﬁrm, the production externality will change from negative to positive for suﬃcient increment of the rival’s output.
This negative-positive production externality may come from the hypothesis of laborhire competition of duopoly firms 1 . Since it can be positive or negative, production
externality determines the total external eﬀects that result in the reaction patterns ofeach ﬁrm.
From (1), (2) and (3) The proﬁt functions of the ﬁrms become accordingly,
Π1 = px − cx (y)x and Π2 = py − cy (x)y.

(4)

Since proﬁt of each ﬁrm is a quadratic polynomial of its own output, solving the ﬁrstorder condition of the proﬁt maximization problem always gives unique solution for agiven value of the rival’s output. This optimal output is known as the reaction function.
Let ri be the reaction function of ﬁrm i, so thatr1 (y e ) = arg max Π1 (x, y e ) and r2 (xe ) = arg max Π1 (xe , y)
x

y

(5)

where xe and y e are output expectations. Substituting (4) into (5), we haver1 (y) = (αy − α + 1)2 ,and r2 (x) = (βx − 1)2 .

(6)

If 1 ≤ α, β ≤ 2 both of r1 and r2 map from I to I. We restrict the combinations of theparameters, α and β , to the following set A:
A ≡ {(α, β)|1 ≤ α, β ≤ 1}

(7)

and assume that a ≥ 3b to keep the marginal costs nonnegative for arbitrary x and y
in I. Due to the presence of total external eﬀects, the reaction function can be upwardor downward-sloping according to whether the external eﬀect is positive or negative.
1

Suppose that ﬁrms demand labor for output production, and there are many people who want towork. If ﬁrm 2 produces only a small amount of output, it employs only a few people as employment
and output production are positively correlated. Thereby, there are many people left unemployed andﬁrm 1 can hire at a lower wage rate. While ﬁrm 2 expands output, unemployed people can increase
their negotiating power so that ﬁrm 1 inevitably oﬀers a higher wage rate to employ more people.
However, if ﬁrm 2 employs most of the people, the remaining unemployed people have less negotiating
power than before and thus ﬁrm 1 can hire them at a lower wage rate again.

3

If α = 1, r1 is monotonically increasing with y. As α becomes larger, the shape of r1
becomes U-shaped. On the contrary, r2 is monotonically decreasing with x with β = 1
and takes U-shaped proﬁle for further increment of β.
About the expectation of each ﬁrm toward the rival’s future output, we make asimple assumption — the so called naive expectation — that is “tomorrow’s expected
output level is today’s observed output level”. Suppose therefore that ﬁrms have naive
expectations, xet = xt−1 and yte = yt−1. The process of output adjustment is
xt+1 = (αyt − α + 1)2 ,yt+1 = (βxt − 1)2 .

(8)Intersections of these two reaction functions are the Cournot-Nash equilibria at
which each ﬁrm’s expectation toward the other ﬁrm’s behavior is conﬁrmed by its
actual behavior. If the expectation is not correct, then ﬁrms adjust their outputs.
We assume the Markov-Perfect-Equilibrium (MPE hereafter) process of adjustment,
which implies that ﬁrms choose their outputs iteratively as the best response to therival’s output in the previous period. Therefore, if the adjustment process of a Cournot
duopoly market follows MPE process, the state of economy (i.e. the output bundle ofduopoly ﬁrms) is always deﬁned in the union of the reaction curves2

3

Equilibria

In this section, we investigate the conditions of existence and stability of equilibrium
points of dynamical system (8). Since (8) consists of asymmetric quadratic equations,
we can not easily identify the explicit conditions for equilibrium analytically. Instead,
we approximate the implicit conditions numerically. Section 3.1 considers the existenceof equilibrium points and Section 3.2 examines the stability and bifurcations of thesepoints.

3.1

Existence of equilibrium points

Since each ri is a quadratic polynomial of the rival’s output, (8) may have four equilibria
S1 , S2 , S3 and S4 . To identify these equilibria, we ﬁrst combine r1 with r2 to deﬁne a
one-dimensional map F :F (x) ≡ r1 ◦ r2 (x) = (1 + αβx(βx − 2))2(9)Here F : I → I, since each reaction function ri : I → I. A ﬁxed point of F in I
corresponds to an equilibrium point of (8). Since it is a forth-order polynomial, F
possibly has four ﬁxed points. Denoting a ﬁxed point of F in I by x∗j (j = 1, . . . , 4),
we assume that x∗1 < x∗2 < x∗3 < x∗4 for convenience. Therefore for Sj (j=1, . . . ,4),
Sj = (x∗j , yj∗ ) such that x∗j = F (x∗j ) and y ∗ = r2 (x∗2 )2(10)

Although we assume that ﬁrms simultaneously update their outputs, the adjustment process
follows MPE process whenever we restrict the initial condition on the union of the reaction curves.

4

To identify the existence of each x∗i , we ﬁrst examine the ﬁxed points of F from
more general viewpoint through following lemma.
F

F

1

1

x∗1

1

x

(a) α = 1.6 and β = 1.7

F

1

1

x∗2

1

x∗4 = 1(b) α = 1.6 and β = 2

F

x∗1

x

x∗1

x

x∗1

(c) α  1.84 and β = 1.33x∗2 x∗31x(d) α = 1.75 and β = 1.9

Figure 1: Determinations of ﬁxed points of F .
Lemma 1 F possesses four fixed points. Moreover,(i) the minimal fixed point of F always exists in the interval (0, 1) for arbitrary
parameter combinations in A and(ii) the maximal fixed point exists in I and is equal to unity if and only if β = 2.
Proof First we set
F  (x) ≡∂F= 4αβ(βx − 1)(αβx2 − 2αβx + 1)
∂x
5

(11)

and
∂2F
= 4αβ 2 (1 + α(3β 2 x2 − 6βx + 2).
(12)∂x2(i) F (0) = 1 for arbitrary parameter combination in A. In addition, F  (x) < 0 andF has no ﬁxed point in (−∞,
0]. Solving
F  (x) > 0 for all x in (−∞, 0]. Therefore
F (x) = 0, we have x = (α ± α(α − 1))/αβ. Let x̂ = (α − α(α − 1))/αβ.
Then 0 < x̂ ≤ 1 since 1 ≤ α, β ≤ 2. This immediately leads to the conclusionthat F (x) always cuts once 45-degree line from above in the interval (0, x̂) andthen this intersection is the minimal ﬁxed point of F .
F  (x) ≡(ii) F (1) = 1 if α = 2/β(2 − β) or β = 2. In the former case, α = 2 with β = 1and α > 2 with 1 < β ≤ 2. Then assume that α = 2 and β = 1. Here x = 1
is a ﬁxed point of F and F  (1) = 0 and F  (1) = −8. However for suﬃciently
large x, both F  (x) F  (x) become positive and this implies that F has at least aﬁxed point in (1, ∞). Considering the latter case, now suppose that β = 2. ThenF  (x) = 8α(2x−1)(4αx(x−1)+1) > 0 and F  (x) = 16α(1+12αx(x−1)+2α) > 0
for all x in [1, ∞). Thus F has no ﬁxed point in (1, ∞), that is, x = 1 is the
maximal ﬁxed point of F if and only if β = 2.
From Theorem 1, we get 0 < x∗ < 1 and x∗ 4 = 1. (See Figure 1(a) and 1(b).)
Considering the existence of x∗2 and x∗3 , let us start from (α, β) = (1, 1). Here onlyx∗1 exists since x̂ = 1. For increment of α and β x∗2 appears in (x̂, 1] if F tangents to
45-degree line from below, see Fig. 1(c). If α or β increases further, then x∗3 appearsin (x∗2 , 1], see Fig. 1(d).
To identify the implicit conditions about the existence of x∗2 and x∗3 deﬁne a mappingφ:(φ(β), β) ∈ A such that F  (x∗2 ) = 1 and F  (x∗2 ) < 0.

(13)

Then A can be divided into the following three subsets A1 , A2 and A3 :A1 ≡ {(α, β) ∈ A | α < φ(β)},
A2 ≡ {(α, β) ∈ A | α = φ(β)},
A3 ≡ A \ (A1 ∪ A2 ).

(14)

Regarding the existence of x∗2 and x∗3 (i.e. S2 and S3 ) we have following states:(i) If (α, β) ∈ A1 , then neither x∗2 nor x∗3 exists,(ii) if (α, β) ∈ A2 , then only x∗2 exists and(iii) if (α, β) ∈ A3 , then both of x∗2 and x∗3 exist.
From Lemma 1, x∗1 always exists in all of the three cases above and x∗4 exists only if
β = 2. These three subsets are illustrated in Figure 2.
To conclude this subsection we state following theorem.
Theorem 1 There exists a subset of A in which all possible fixed points of F exist in
I. Therefore the adjustment process of (8) has four distinct equilibria.
6

2

A2

Beta

1.8

A3

1.6
1.4

A1
1.2

1.2

1.4 1.6
Alpha

1.8

2

Figure 2: Existence of the equilibrium points.
Proof F  (x) = 0 if x = β1 and then F ( β1 ) = (1 − α)2 and F  ( β1 ) = 4α(1 − α)β 2 ≤ 0.
Solving (1 − α)2 ≥ β1 , we have1
α <1− √
β1or α > 1 + √ .
β(15)Therefore for arbitrary β in [1, 2], x∗2 and x∗3 appear whenever α increases toward 2.
From Lemma 1, x∗1 always exists and x∗4 exists if β = 2. Thus if α > 1 + √12 and β = 2,then F has four ﬁxed points in I, that is, all of S1 , S2 , S3 and S4 exist.

3.2

Stabilities and bifurcations of equilibrium points

To identify the stability condition of each equilibrium point we linearize the system (8)around the equilibria. Suppose that J is the Jacobian matrix of (8) such that
⎞
⎛
dr1
0
⎜
dy ⎟
(16)
J ≡⎝
⎠
dr2
0
dx
with eigenvalues λ1 = λ2 = λ. Then an equilibrium point Si is stable if and only if dr1 dr2 ∗ ∗ (x , y ) = |F  (x∗j )| < 1(17)|λ| ≡ dy dx j j 
where the derivatives are evaluated at Sj .
Here we use again the one-dimensional map F for checking the stabilities of theequilibrium points. Si = (x∗j , r2 (x∗j )) is a stable equilibrium point if and only if x∗j is astable ﬁxed point of F . From Figure 1 we can see that x∗2 and x∗4 (i.e. S2 and S4 ) areunstable whenever they exist. In addition, it can be numerically checked that x∗1 (i.e.
S1 ) is unstable for arbitrary parameter combinations in A. Therefore we only examine
the stability of x∗3 (i.e. S3 ).
7

It is known that a dynamical system with logistics-like maps has ﬂip bifurcations
when it loses stability. That is, for the system in this paper, (8) loses stability if
λ ≤ −1. Here we deﬁne a mapping ψ:(ψ(β), β) ∈ A such that F  (x∗3 ) = −1.

(18)

Now A can be divided into following two subsets, B1 and B2 :B1 ≡ {(α, β) ∈ A | α < ψ(β)} and B2 ≡ A \ B1 .

(19)

Regarding the stability condition of x∗3 (i.e. S3 ) we have the following states:(i) If (α, β) ∈ B1 ∩ A3 , then S3 is stable and(ii) if (α, β) ∈ B2 , then S3 is unstable.

2

Beta

B2

A3 ∩ B1

1.8
1.6
1.4

A1
1.2

1.2

1.4 1.6
Alpha

1.8

2

(a) stability condition of x∗3(b) bifurcation diagram

Figure 3: Stability and bifurcation of the equilibrium points.
In Figure 3 we calculate the implicit stability condition of x∗3 (Fig. 3(a)) and thebifurcation diagram of F with respect to α and β (Fig. 3(b)). A dark-colored belt in
Fig. 3(b) corresponds to B1 ∩ A3 in Fig. 3(a). In white-colored regions in Fig. 3(b),the dynamics generates non-periodic ﬂuctuations, that is, MPE process of output may
be chaotic.

4

Statistical Dynamics

In this section we investigate the long-run statistical behavior of the system (8). Since
analytical investigations are diﬃcult or not possible, numerical experiments are used8

to calculate the proﬁts taken at the equilibriums (equilibrium proﬁts hereafter) andthe long-run average proﬁts taken along MPE process (average proﬁts hereafter). We
compare the equilibrium proﬁts and the average proﬁts and then check that whether
ﬁrms can eventually beneﬁt from chaotic ﬂuctuations instead of settling at possible
equilibria. In Section 4.1 we focus on the case in which α = β = 2. In Section 4.2 weextend the calculation to entire region of A.

4.1

Ergodic Chaos

Suppose that α = β = 2. Now the system (8) consists of symmetric reaction functions
r1 (y) = (2y − 1)2 and r2 (x) = (2x − 1)2 and has four equilibrium points, so thatequivalently, F has four ﬁxed points in I such that√√3− 5 ∗ 1 ∗ 3+ 5∗
, x2 = , x3 =and x∗4 = 1.
(20)
x1 =
8
4
8
We evaluate proﬁts of duopoly ﬁrms at each equilibrium point Sj = (x∗j , r2 (x∗j )). Theseare√( 5 − 3)2 b
1
3
Π1 = Π2 =,64b
Π21 = Π22 = ,
(21)
16
√
2
(
5
+
3)
b
and
Π31 = Π12 =64
Π41 = Π42 = b.
where Πji denotes equilibrium proﬁt of ﬁrm i at Sj .
It is known that if α = β = 2, the chaotic ﬂuctuation generated by the adjustment
process is ergodic. From M-N [10], if chaos is ergodic, the average proﬁts with MPEprocess can be analytically derived. Let Π̄i be the average proﬁt of ﬁrm 1 with MPEprocess, then
T −1

Π̄i

1
= lim
Πi (xt , yt )T →∞ T
i=0
1
2
b
=
4

=

1

0

Πi (r1 (u), u)Φ(u)du +10Πi (u, r2 (u))Φ(u)du
(22)

where
1
Φ(u) = .
π u(1 − u)

9

(23)

Constructing a proﬁt ordering of each ﬁrm, we have
Π11 < Π21 < Π̄1 < Π31 < Π41(24)

Π32 < Π22 < Π̄2 < Π12 < Π42 .

Therefore duopoly ﬁrms cannot always beneﬁt from chaotic ﬂuctuations when α = β =2.

4.2

Beneficial Chaos

Now we extend the analysis to entire region of A. From Fig. 3(b), we can see thatdynamical system (8) generates chaotic ﬂuctuations although neither α nor β equals
2. However, since ergodicity does not hold in these cases, the average proﬁts generated
by chaotic ﬂuctuations with these parameter combinations can not be bounded bytheoretically derived values. Instead, we check the actual value of the average proﬁts
with suﬃciently large number of periods. For each combination of the parameters we
calculate possible equilibrium proﬁts and the average proﬁt under chaos and then checkthe proﬁt ordering of each ﬁrm.

(a) ﬁrm 1(b) ﬁrm 2

Figure 4: Variation of the proﬁt order.
Figure 4 illustrates the variation of the proﬁt ordering of each ﬁrm3 . The colorsof Fig.4 correspond to the preference ordering of ﬁrms about the average proﬁt underchaos. That is, each color depicts how many equilibrium points exist, at which the
proﬁt is larger than the average proﬁt. In white-colored regions no equilibrium proﬁt
3

For each parameter combination, we calculate the average proﬁt for 1000 periods after a 2000period iterations.

10

is larger than the average chaotic proﬁt (i.e. Π̄i ≥ Πji > · · · ). As the color gets darker,
the preference ordering about the average proﬁt becomes worse. In the black-colored
region, therefore, the average chaotic proﬁt is less than all possible equilibrium proﬁts(i.e. Π̄i < Πji < · · · ).
Comparing Fig. 4(a) and Fig. 4(b) we can say that there exists a subset of A in
which, for both ﬁrms, the average chaotic proﬁts are larger than all possible equilibrium
proﬁts. Since this subset is in A1 , only S1 exists and we have the proﬁt ordering:Π11 < Π̄1and Π12 < Π̄2 .

x

(25)

Π

1
0.8
0.6

0.3

Π̄2

0.2

Π12
Π̄1

0.4

0.1
0.2

1.6

1.7

1.8

1.9

Π11

β
2

1.6

(a) bifurcation cascade

1.7

1.8

1.9

2

β

(b) variation of the average proﬁts

Figure 5: An example of beneﬁcial chaos (α = 1.1).
An example of beneﬁcial chaos is illustrated in Figure 5, where α is ﬁxed at ᾱ = 1.1
and β varies (1.5 ≤ β ≤ 2). As β increases MPE process becomes chaotic, (see Fig.
5(a)) and then the average proﬁts of both ﬁrms becomes larger than the equilibrium
proﬁts at S1 , see Fig. 5(b).

5Concluding Remarks

This paper presents two main results. Firstly, a Cournot duopoly market may bechaotic if ﬁrms have U-shaped reaction functions. To construct an economic foundation of U-shaped reaction functions, the investigation takes account of two patterns of
nonlinear, negative-positive production externalities. Secondly, heterogeneous duopoly
ﬁrms can eventually beneﬁt from chaotic adjustment of output when production externalities are asymmetric. To demonstrate beneﬁcial chaos, some numerical experiments
calculate the long-run average proﬁts taken along chaotic trajectories and then compare them with proﬁts taken at possible equilibrium points. The comparison showsthat there exists a subset of parameter combinations for which, and for both the ﬁrms,
the long-run average proﬁts are strictly larger than equilibrium proﬁts. In that sense,
we can conclude that chaotic ﬂuctuation can be beneﬁcial in a heterogeneous market.
11Since chaotic ﬂuctuations do not converge to an equilibrium state, they are thoughtof as an unfavorable phenomenon in traditional economics. However, the results obtained in this paper, in particular, the last numerical result, throws an interesting lighton the nature of complex chaotic dynamics and suggests a further study on duopoly
dynamics involving chaos worthy of consideration.

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