Dynamics in Delay Cournot Duopoly∗
Akio Matsumoto†Chuo University

Ferenc Szidarovszky‡
University of Arizona

Hiroyuki Yoshida§
Nihon University

Abstract
Dynamic linear oligopolies are examined with continuous time scales
and information delays. Systems dynamics are compared with fixed andcontinuously distributed time lags. We first show that by assuming constant speeds of adjustment stability is preserved if the firms use instantaneous information on their outputs and have delays in the outputs of thecompetitors, and the stability might be lost if time delays are present in
the firms’ own outputs. Similar results are obtained if each firm adjusts
its growth rate proportionally to a change in its profit. In the case ofstability loss Hopf bifurcation occurs giving the probability of the birth oflimit cycles around the stationary state.

1

Introduction

Oligopoly models play a central role in the literature of mathematical economics.
Since the pioneering work of Cournot (1838), a large number of researchers discussed and examined the classical Cournot model and its variants and extensions. The existence and uniqueness of the equilibrium was the main focus ofstudies in the early stages and later the research turned to the dynamic analysisof oligopolistic markets. A comprehensive summary of earlier results can befound in Okuguchi (1976), and their multiproduct generalization with several
case studies are discussed in Okuguchi and Szidarovszky (1999). The attention
was focused on linear and linearized models at the beginning which provided
the local asymptotic properties of the equilibria. During the last two decades
however an increasing attention has been given to the analysis of global dynamics. A survey of the newer results can be found in Bischi et al. (2009) which∗ 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, 742-1, Higasi-Nakano, Hachioji, Tokyo, 192-0393, Japan.
akiom@tamacc.chuo-u.ac.jp‡ Department of Systems and Industrial Engineering, Tucson, Arizona, 85721-0020, USA.
szidar@sie.arizona.edu§ Collegeof Economics, 1-3-2, Misaki, Chiyoda, Tokyo, 101-8360, Japan.
yoshida.hiroyuki@nihon-u.ac.jp

1contains models with both discrete and continuous time scales. The most common dynamic processes are based either on the gradients of the profit functions
or on best responses. In both cases usually constant speeds of adjustment are
assumed and very few studies consider output-dependent speeds of adjustment.
In the case of discrete time scales we mention the works of Bischi and Naimuzada(1999) and Bischi and Lamantia (2002). In this paper continuous time scales
will be considered.
In the cases of most models discussed earlier in the literature, it was assumed
that each firm has instantaneous information about its own output and also on
the outputs of the competitors. This assumption has mathematical convenience,
however it is unrealistic in real economics, since there are always time delays
due to determining and implementing decisions. In addition to these facts, infast changing industries the firms do not want to follow sudden market changes,
they rather want to react to averaged past information. Hence there are always
time delays between the times when information is obtained and the times when
the decisions are implemented.
Howroyd and Russel (1984) constructed a linear continuous dynamic oligopoly
model in which the outputs were adaptively adjusted and the adjustments were
subject to fixed time delays. Their conclusions were very clear: stability is not
aﬀected by the information lags about the rivals’ outputs while it is aﬀected by
the information lags about the firms’ own outputs. There are many economic
situations in which the lags are uncertain or the firms are reacting to averaged
past information. In such situations, continuously distributed time lags are useful. In this study, we have two main purposes: we first examine whether or
not the Howroyd-Russel results still hold in a dynamic oligopoly model with
continuously distributed time lags; second, we reconstruct the model such thatthe growth rate of the outputs are adjusted with the gradient method and show
the existence of complex dynamics when a stationary state loses stability.
The paper is organized as follows. Section 2 constructs a basic duopoly
model with linear price and cost functions. Section 3 assumes constant speedof adjustment and introduces information time delays into the basic model.
Section 4 discusses a nonlinear extension of the basic model by adopting the
growth rate adjustment process. Section 5 concludes the paper.

2

The Basic Model

Dynamics in a classical duopoly model is considered. Let x and y be the quantities produced by firm x and firm y with linear production costs where themarginal costs are denoted by cx and cy . The price function is also assumed tobe linear,p = a − b(x + y) with a > 0 and b > 0.
The profit functions are given byπx (x, y) = (a − b(x + y)) x − cx x2andπ y (x, y) = (a − b(x + y)) y − cy y.
Firm x determines its output to maximize its profit with respect to x and sodoes firm y with respect to y. Assuming interior optimal points and solving thefirst-order conditions for the outputs yield the best reply function for firm x,
a − cx − by2b

Rx (y) =and the best reply function for firm y,
Ry (x) =a − cy − bx.
2b

A Cournot point is an intersection of these best reply functions and its coordinates area − 2cx + cy
xc =
3b
and
a − 2cy + cx
yc =
.
3b
In this paper we assume that the firms continuously adjust their outputs proportionally to the change in their profits (i.e., gradient dynamics),ẋ = α(x)

∂π y
∂π x
and ẏ = β(y)
∂x
∂y

(1)

where α(x) and β(y) are positive adjustment functions of firm x and firm y.

3

Delay Linear Duopolies

In this section, we start with a simple case and assume constant adjustment
coeﬃcients:
Assumption 1. α(x) = α > 0 and β(y) = β > 0.
The continuous dynamic duopoly model is⎧⎨ ẋ(t) = α(a − cx − 2bx(t) − by(t))⎩(2)ẏ(t) = β(a − cy − bx(t) − 2by(t)).

Using the best reply functions, the dynamic system can be rewritten as⎧⎨ ẋ = ᾱ (Rx (y) − x)⎩ẏ = β̄ (Ry (x) − y)3(3)where ᾱ = 2bα and β̄ = 2bβ. In system (3), each firm adaptively adjusts its
output in such a way that the adjustment rate of the output is proportional
to the diﬀerence between the profit maximizing output and the current output.
That is, each firm adjusts its output into the direction toward its best reply.
The transformation from (2) to (3) or vice versa implies that for the firms, thegradient adjustment of the output is the same as the adaptive adjustment toward
best reply. To examine the stability of system (2), we consider its coeﬃcientmatrix,
⎛
⎞
−2bα
−bα
⎠
J =⎝
−bβ
−2bβ
with trace

trJ = −2b(α + β) < 0
and determinantdetJ = 3b2 αβ > 0.
Since (2) is linear, and the above conditions imply local asymptotical stability,
we confirm the following well-known result:
Theorem 1 The continuous dynamic duopoly model (2) is globally asymptotically stable.
Howroyd and Russel (1986) introduced the fixed delay adjustment process
in a general n-firm oligopoly model and showed two main conclusions: first, theinformation delays in the competitors’ outputs are harmless to the stability ofthe models and second, stability is aﬀected if all information available to thefirms is subject to a delay. A duopoly version of their model is presented by⎧⎨ ẋ(t) = α(a − cx − 2bx(t − Sx ) − by(t − Tx ))(4)⎩ẏ(t) = β(a − cy − bx(t − Ty ) − 2by(t − Sy ))where the firms adjust their current outputs based on delayed information atsome preceding times t − Si and t − Ti for i = x, y. They consider the situation
based on information in which each firm experiences a time lag Ti in obtaining information about the rival’s output and a time lag Si in implementing
information about its own output.
The characteristic equation of system (4) can be obtained by looking for thesolutions asx(t) = eλt u and y(t) = eλt v
and substituting them into the corresponding homogeneous equations:⎧⎨ λeλt u = α(−2beλ(t−Sx ) u − beλ(t−Tx ) v),⎩

λeλt v = β(−beλ(t−Ty ) u − 2beλ(t−Sy ) v).
4

Nontrivial solution exists if and only if the determinant of the coeﬃcient matrix⎛⎞
λ + 2bαe−λSxbαe−λTx⎠
JF = ⎝
(5)
−λTy
−λSy
bβe
λ + 2bβe
is zero, which provides a mixed exponential-polynomial equation for λ:¡¢¡¢λ + 2bαe−λSx λ + 2bβe−λSy − b2 αβe−λTx e−λTy = 0.

Multiplying both sides by eλSx eλSy eλTx eλTy , we get
¡ λSx
¢¡
¢
λe
+ 2bα λeλSy + 2bβ eλTx eλTy − b2 αβeλSx eλSy = 0.

(6)

To clarify the eﬀects caused by the time lags on dynamics, we first examine
the case in which the firms adjust their outputs using correct information on
their own outputs and have uncertainty about their rival’s output (i.e., Sx =Sy = 0 while Tx > 0 and Ty > 0) and then consider the opposite case in which
the firms have lags on their outputs (i.e., Tx = Ty = 0 while Sx > 0 and Sy > 0).
Substituting Sx = Sy = 0, Tx > 0 and Ty > 0 into (6), we obtain thecharacteristic equation(λ + 2bα)(λ + 2bβ) − b2 αβe−λ(Tx +Ty ) = 0.
Notice first that λ = 0 cannot be a solution. So assume that λ 6= 0 and R(λ) ≥ 0
where λ = μ + iξ. Then we have|λ + 2bα| |λ + 2bβ| ≥ 4b2 αβ
and

¯
¯
¯¯
¯
¯
¯ 2
¯
¯¯
¯
¯
¯b αβe−λ(Tx +Ty ) ¯ = b2 αβ ¯e−μ(Tx +Ty ) ¯ ¯e−iξ(Tx +Ty ) ¯ ≤ b2 αβThe last two inequalities imply that no λ with R(λ) ≥ 0 can solve the characteristic equation. Hence any solution of the characteristic equation must havenegative real parts. The information lags on the rival’s outputs are harmlessand do not aﬀect the stability of (4).
Taking Tx = Ty = 0, Sx > 0 and Sy > 0 and repeating the same procedureyield the characteristic equation of the form
λ2 − b2 αβ + 2bαλe−λSx + 2bβλe−λSy + 4b2 αβe−λ(Sx +Sy ) = 0.

(7)

In the absence of information lags, the linear system (4) is locally asymptoticallystable. The characteristic equation implies that λ = 0 is not a solution of (7).
By continuity, all eigenvalues of (7) have negative real parts for suﬃciently
small Sx + Sy > 0. Freedman and Rao (1986) estimated the range of Sx + Sy forwhich the Cournot point remains asymptotically stable. In particular, according5

to their theorem (i.e., (ivβ) of Theorem 3.1), the following estimates on Sx andSy imply asymptotical stability:
max[Sx , Sy ] ≤ Γ,
Sx + Sy <η
,
4b2 αβCos[Γv+ ] =η,2b(α + β)

where
0 < η < 2b(α + β)
and

o
p
bn
2(α + β) + (α + β)2 + 12αβ .
2Figure 1 illustrates two time trajectories of x when α = β = 1, b = 1 and η = 4.
Under these parameter specifications, the Cournot point is asymptotically stable
for max[Sx , Sy ] ≤ 0.262 and Sx + Sy < 1. We take Sx = 0.24 and Sy = 0.26
in Figure 1(A) and Sx = 0.7 and Sy = 0.8 in Figure 1(B). It can be seen thatthe large information lags on the own outputs of the firms have an instabilizing
eﬀect. We thus have confirmed the results of Howroyd and Russel (1986) in the
duopoly case.
v+ =(A) Converging trajectory(B) Diverging trajectory

Figure 1. Time trajectories of x(t) generated by (4)In real economic situations, the lags are usually uncertain and can be considered to be fixed only under special circumstances. Therefore, we will model timelags in a continuously distributed manner. Given Assumption 1, the continuous
dynamic duopoly model with continuously distributed time lags is⎧⎨ ẋ(t) = α(a − cx − 2bxε (t) − by e (t)),(8)⎩ẏ(t) = β(a − cy − bxe (t) − 2by ε (t)),
6with expectations on its own outputs
Z t
Z t
ε
ε
ω(t − s, Sx , nx )x(s)ds and y (t) =ω(t − s, Sy , ny )y(s)ds
x (t) =
00

and expectations on its rival’s outputs
Z t
Z tω(t − s, Tx , mx )x(s)ds and y e (t) =ω(t − s, Ty , my )y(s)ds.
xe (t) =00

The weighting function ω is assumed to have the form
⎧ 1 t−s
⎪
e− Γ
⎪
⎪
⎨ Γ
ω(t − s, Γ, `) =µ ¶`+1⎪⎪`(t−s)⎪ 1 `⎩(t − s)` e− Γ`! Γ

if ` = 0,(9)
if ` ≥ 1.

Here we assume that Γ > 0 and ` is a nonnegative integer. Substituting (9) into
the expectation formations defined above, and the resulting expressions of the
expectations into (8) yield a system of integro-diﬀerential equations. In orderto analyze the dynamic behavior of the system, we consider the corresponding
homogeneous system. Letting xδ and yδ denote the deviations of x and y fromtheir Cournot levels of outputs, xc and y c , the homogeneous system can beformulated as follows:
⎧
n
o
Rt
Rt
⎪
ẋ
=
α
−2b
ω(t
−
s,
S
,
n
)x
(s)ds
−
b
ω(t
−
s,
T
,
m
)y
(s)ds
,
⎪
δ
x
x
δ
x
x
δ
⎨
0
0
n R
o
⎪
R
⎪
⎩ ẏδ = β −b t ω(t − s, Ty , my )xδ (s)ds − 2b t ω(t − s, Sy , ny )yδ (s)ds .
0

0

(10)

We seek the solutions in the exponential formxδ (t) = eλt u and yδ (t) = eλt v.
Substituting these solutions into equation (10) and arranging terms yield
³
´
Rt
Rt
λ + 2bα 0 ω(t − s, Sx , nx )e−λ(t−s) ds u + bα 0 ω(t − s, Tx , mx )e−λ(t−s) ds · v = 0,
³
´
Rt
Rt
bβ 0 ω(t − s, Ty , my )e−λ(t−s) ds · u + λ + 2bβ 0 ω(t − s, Sy , ny )e−λ(t−s) ds v = 0.

Introducing a new variable z = t − s, we can simplify the integral terms bynoticing thatZ tZ t−λ(t−s)ω(t − s, Γ, `)eds =ω(z, Γ, `)e−λz dz.
00Allowing t → ∞, we haveµ¶−(`+1)Z tλΓ−λz
ω(z, Γ, `)edz = 1 +
lim
t→∞ 0
q
7

with
q=

⎧
⎨ 1 if ` = 0,⎩`

if ` ≥ 1.

Then equations (10) can be simplified as⎧ Ãµµ¶−(nx +1) !
¶−(mx +1)
⎪
λS
λTx
⎪
x
⎪
λ + 2bα 1 +v=0u + bα 1 +
⎪
⎪
⎪
qx
q̄x
⎨

Ã
⎪
µ
µ
¶−(my +1)¶−(ny +1) !
⎪
⎪
λT
λS
⎪
y
y
⎪
u + λ + 2bβ 1 +v = 0.
⎪⎩ bβ 1 + q̄yqywhich can be rewritten in the matrix form:
µ
¶µ
¶ µ ¶
Ax (λ) Bx (λ)
u0=By (λ) Ay (λ)
v
0
where

" µ
#µ
¶n +1
¶mx +1
λSx x
λTx
Ax (λ) = λ 1 ++ 2bα 1 +
,
qx
q̄x
" µ
#µ
¶n +1
¶my +1
λSy y
λTy
Ay (λ) = λ 1 +
+ 2bβ
1+
,
qy
q̄y
µ
¶n +1
λSx x
Bx (λ) = ba 1 +
qx

and

µ
¶n +1
λSy y
By (λ) = bβ 1 +.
qy

A non-trivial solution exists if and only if
Ax (λ)Ay (λ) − Bx (λ)By (λ) = 0.

(11)

We denote the left hand side by ϕ(λ) and call it the characteristic polynomialof system (10).
There is an interesting relation between the characteristic polynomials (6)and (11) of the systems with fixed and continuously distributed time lags. Assume that nx , ny , mx and my converge to infinity. Since qi = ni − 1 andq̄i = mi − 1 for i = x, y, we have the limits¡¢
Ax (λ) → λeλSx + 2bα eλTx ,¡¢Ay (λ) → λeλSy + 2bβ eλTy ,Bx (λ) → bαeλSx8andBy (λ) → bβeλSy ,therefore in the limiting case, the characteristic equation of the continuously
distributed delay model converges to the characteristic equation of the fixed
delay model.

3.1

Information lags about the rival’s output

In this section we confine our analysis to the case in which firms experience no
information lags on their own outputs:
Assumption 2. Sx = 0 and Sy = 0.
Under Assumption 2, the characteristic equation has the form¶mx +1 µ¶my +1µ
λTy
λTx− b2 αβ = 01+(λ + 2bα)(λ + 2bβ) 1 +
q̄x
q̄y

(12)

We will prove that all roots of equation (12) have negative real parts implying
the global asymptotical stability of the Cournot point. Assume in contrary that
there is a root λ = α + iβ with α ≥ 0. Notice first that with any real positive
numbers A and B,
2|A + Bλ|

=
=
≥
≥

2

|(A + Bα) + i(Bβ)|(A + Bα)2 + (Bβ)2(A + Bα)2A2 .

Therefore the two terms of (12) have to be diﬀerent, since¯µ¶mx +1 µ¶my +1 ¯¯
¯
λTx
λTy
¯
¯
1+
¯(λ + 2bα)(λ + 2bβ) 1 +¯ ≥ 4b2 αβ > b2 αβ.
¯
¯
q̄x
q̄y

Consequently λ cannot satisfy equation (12). This observation implies that thestability part of the result of Howroyd and Russel (1986) remains true in thecase of continuously distributed lags.

3.2

Information lags on the own outputs

In this section we confine our analysis to the case in which the firms have no
information lags on their rivals’ outputs.
Assumption 3. Tx = 0 and Ty = 0.

9Under Assumption 3, the characteristic equation has the form
Ã µ
!Ã µ
!
µ
¶n +1
¶n +1
¶n +1 µ
¶n +1
λSx x
λSy y
λSx x
λSy y
2
λ 1+
+ 2bα
λ 1+
+ 2bβ −b αβ 1 +
=0
1+
qx
qy
qx
qy
(13)
Since it is diﬃcult to check whether the real parts of the roots are negative or
positive in general, we will show some simple special cases in which analyticalresults can be obtained.
Case 1. Sx > 0 with nx = 0 and Sy = 0.
As a special case, we assume first that firm x has information lag on its own
output, however, firm y uses instantaneous information about its own output,that is, Sx > 0 and Sy = 0. We also assume exponential weighting function
with nx = 0. Then equation (13) reduces to the following cubic equation:
a0 λ3 + a1 λ2 + a2 λ + a3 = 0
where the coeﬃcients are defined as
a0 = Sx > 0,
a1 = 1 + 2bβSx > 0,
a2 = 2b(α + β) − b2 αβSx R 0,
a3 = 3b2 αβ > 0.
Furthermore, we have©
ªa1 a2 − a0 a3 = 2b −b2 αβ 2 Sx2 + 2bβ 2 Sx + (α + β) .

(14)

If a2 > 0 and a1 a2 − a0 a3 > 0 are confirmed, then the Routh-Hurwitz stability
theorem implies that the system (10) is globally asymptotically stable. It suﬃces
for our purpose to show that a1 a2 − a0 a3 > 0 in this case since a0 > 0, a1 > 0
and a3 > 0 imply that a2 > 0. Let f (Sx ) be the right hand side of (14). Sincef (0) > 0 and f 0 (0) > 0, equation f (Sx ) = 0 has one positive root,p
β + α2 + αβ + β 2
0
Sx =
> 0.
bαβ
Hence we have a2 > 0 and a1 a2 − a0 a3 > 0 for Sx < Sx0 . We summarize thisresult as follows:
Theorem 2 If Sx > 0 with nx = 0 and Sy = 0 under Assumption 3, then the
delay dynamic system (8) is globally asymptotically stable when 0 < Sx < Sx0
and it is unstable when Sx > Sx0 wherep
β + α2 + αβ + β 2
0
Sx =
.
bαβ
10

Case 2. Sx > 0 with nx = 0 and Sy > 0 with ny = 0.
In the second special case, we assume that both firms have information lags
about their own outputs and have exponentially declining weighting functions.
Then equation (12) reduces to the following fourth degree polynomial equation:
a0 λ4 + a1 λ3 + a2 λ2 + a3 λ + a4 = 0
where the coeﬃcients are defined as
a0 = Sx Sy > 0,
a1 = Sx + Sy > 0,
a2 = 1 + 2b(αSy + βSx ) − b2 αβSx Sy R 0,
a3 = 2b(α + β) − b2 αβ(Sx + Sy ) R 0,
a4 = 3b2 αβ > 0.
The Routh-Hurwitz stability theorem implies that the roots of the characteristic
equation have negative real parts if and only if all coeﬃcients are positive andthe following determinants are positive:¯¯¯¯¯ a1 a0 0 ¯¯¯ a1 a0 ¯¯¯ > 0 and J3 = ¯ a3 a2 a1 ¯ > 0.
J2 = ¯¯¯¯¯a3 a2¯ 0 a4 a3 ¯
Notice that

J2 = Sx + Sy + 2b(αSy2 + βSx2 ) > 0

and
J3 = a3 J2 − a21 a4 .
We can easily show that the single condition J3 > 0 implies that all roots havenegative real parts. Since J2 and a1 and a4 are positive, J3 > 0 implies a3 > 0.
Then relation J2 = a1 a2 − a0 a3 > 0 implies that a2 also have to be positive.
Theorem 3 Given Assumption 2, if Sx > 0 with nx = 0 and Sy > 0 with
ny = 0, then the delay dynamic system (8) is globally asymptotically stable for(Sx , Sy ) below the partition line, J3 = 0, and unstable for (Sx , Sy ) above the line
where
ª
©£
¤
J3 = b Sx + Sy + 2b(αSy2 + βSx2 ) [2(α + β) − bαβ(Sx + Sy )] − 3bαβ(Sx + Sy )2

It is possible to obtain the analytic form of the partition line with larger
values of nx and ny . It becomes, however, much more complicated as shown in
Theorem 3. Therefore we check numerically the shapes of the partition lines.
Taking α = β = 1, b = 1 and repeating the above procedure with increasing
values of nx = ny = 1, 2, 3, 4, we obtain the four partition lines illustrated
in Figure 2 in which Pi means the partition line when nx = ny = i. Since
the partition line divides the parameter space into the stable and the unstableregions, stability of the dynamic system (4) is clearly aﬀected by the information11

lags about the firms’s own outputs, similarly to the case of fixed time lags asshown by Howroyd and Russel (1986). Figure 2 indicates that the stable region
becomes larger as the partition line shifts upward with increasing values of nx
and ny and converge to the stable region with fixed time delay when nx and ny
tend to infinity.

Figure 2. Partition lines

4

Delay Nonlinear Duopolies

In this section we consider the case in which the adjustment speeds are positive functions of the outputs of the firms. In particular, we adopt the linear
dependency:
Assumption 4.

α(x) = αx with α > 0 and β(y) = βy with β > 0.

The continuous dynamic system (1) under Assumption 4 can be written as
⎧
ẋ(t)
⎪
⎪
= α(a − cx − 2bx(t) − by(t)),⎪⎪⎨ x(t)
(15)
⎪
⎪
ẏ(t)
⎪
⎪
= β(a − cy − 2by(t) − bx(t)).
⎩y(t)

This implies that each firm adjusts its growth rate of the outputs proportionallyto a change in the profit. As we have done before, system (15) can be also
expressed in terms of the best reply functions,
⎧
ẋ(t)
⎪
⎪
= ᾱ(Rx (y(t)) − y(t)),⎪⎪⎨ x(t)
(16)
⎪
⎪
ẏ(t)
⎪
⎪
= β̄(Ry (x(t)) − x(t)),
⎩
y(t)
12

which can be interpreted as each firm adjusts its growth rate proportionallyto the diﬀerence between its profit maximizing output and its actual output.
The discrete versions of the duopoly dynamic model (15) or (16) are consideredby Huang (2002) adopting a feedback controlling method to stabilize a discrete
system and also by Hassen (2004) showing that the stability region of the Nashequilibrium can become larger in a delayed duopoly model.
As in the case of linear duopoly, we first introduce fixed information lags on
the own behavior and rewrite (15) as⎧⎨ ẋ(t) = αx(t) (a − cx − 2bx(t − Sx ) − by(t)) ,⎩ẏ(t) = βy(t) (a − cy − bx(t) − 2by(t − Sy ))which has the same dynamic structure as the Lotka-Volterra type models. Applying the results obtained by Shibata and Saito (1980), we can demonstratethat this fixed delay Cournot model displays various dynamic behavior ranging
from periodic solutions to chaotic solutions. Figure 3 illustrates emergence of
chaotic oscillation under α = β = 1, cx = cy = 1, a = 3, Sx = 1.6 and Sy = 0.9.

Figure 3. Chaotic solution of the fixed delay Cournot model
We now introduce continuously distributed lags into (15),⎧⎨ ẋ(t) = αx(t) (a − cx − 2bxε (t) − by e (t)) ,⎩(17)ẏ(t) = βy(t) (a − cy − bxe (t) − 2by ε (t)) ,where the expected values of the firms’ own outputs, the expected values of thecompetitors’ outputs and their weighting functions are given as in the previous
section.
To examine local dynamics of the above system in a neighborhood of theequilibrium point (xc , y c ) where xc = xe = xε and y c = y e = y ε , we need to examine the linearized version of system (17). By considering the right hand sides
13

as three-variable functions (depending on x, xε , y e and y, xe , y ε , respectively),
the linearized system becomes
¶
µ
Z t
Z t
w(t − s, Sx , nx )xδ (s)ds − bw(t − s, Ty , my )yδ (s)ds ,ẋδ (t) = αxc −2b
0

0

¶
µ Z t
Z t
ẏδ (t) = βy c −bw(t − s, Tx , mx )xδ (s)ds − 2bw(t − s, Sy , ny )yδ (s)ds ,00which has exactly the same form as the linear system (10) if α and β are replacedby αxc and βy c . Thus the characteristic equation is given byĀx (λ)Āy (λ) − B̄x (λ)B̄y (λ) = 0
where

Ã µ
! Ãµ
¶n +1
¶mx +1 !
λSx x
λTx
c
+ 2bαx
1+,Āx (λ) = λ 1 +
qx
q̄x
Ã µ
! Ãµ
¶n +1
¶my +1 !
λSy y
λTy
c
+ 2bβy1+,Āy (λ) = λ 1 +
qy
q̄y
c

B̄x (λ) = bαx

µ

λSx
1+
qx

¶nx +1

,

µ
¶n +1
λSy y
B̄y (λ) = bβy c 1 +
.
qy

4.1

Eﬀects of the information lag about the own output

Following the same procedure as in the previous section, we have the following
result:Theorem 4 If Sx > 0 with nx = 0 and Sy = 0 under Assumption 3, then the
delay dynamic system (17) is locally asymptotically stable when 0 < Sx < Sx0
and it is locally unstable when Sx > Sx0 wherep
βy c + (αxc )2 + (βy c )2 + αβxc y c0Sx =.
bαβxc y c
The a1 a2 − a0 a3 = 0 line divides the parameter space into two parts: one inwhich the system is locally stable and the other in which it is locally unstable.
We call it the partition line. Substituting a3 = aa1 a0 2 into the characteristic
equation ϕ(λ) = 0 and factorizing it yield (a1 + a0 λ)(a2 + a0 λ2 ) = 0 which canbe solved for λ. Two of the characteristic roots are purely imaginary and thethird is real and negative:ra2
λ1,2 = ± − = ±iξ
a0
14

and
λ3 = −a1< 0.
a0

We show the appearance of Hopf bifurcation giving the possibility of the birthof limit cycles. To this end, we need to confirm whether the real part of thecomplex roots is sensitive to a change in a bifurcation parameter. We select Sx
as the bifurcation parameter and suppose that λ is a function of Sx . Implicitly
diﬀerentiating the characteristic equation gives
λ3 + 2bβy c λ2 + (−b2 αβxc y c )λ
dλ
=
.
dSx
−(3a0 λ2 + 2a1 λ + a2 )Substituting λ = iξ with ξ 2 = aa20 , rationalizing the right hand side and noticingthat the terms with λ and λ3 are imaginary and the constant and quadratic
terms are real yield the following form of the real part of the derivative of λwith respect to the bifurcation parameter:¤£µ¶a2 4(bβy c )2 (bαxc Sx0 − 1)
dλ
Re
=
> 0.
dSx
2a0 (a1 a3 + a22 )We summarize this result as follows:
Theorem 5 The stationary point (xc , y c ) of delay system (17) is destabilized
via a Hopf bifurcation when Sx increases from Sx0 .
Figure 4 shows the birth of a limit cycle in this case. In the opposite case inwhich Sy = 0 with ny = 0 and Sx = 0, we have the same results as Theorems 4
and 5 if Sx , xc and α are replaced by Sy , y c and β.

Figure 4. Birth of a limit cycle
Applying Theorem 3, we also have the following result:
15

Theorem 6 Given Assumption 3, if Sx > 0 with nx = 0 and Sy > 0 with
ny = 0, then the delay dynamic system (17) is locally asymptotically stable for(Sx , Sy ) below the partition line, J3 = 0, and locally unstable for (Sx , Sy ) above
it where£¤
J3 = b Sx + Sy + 2b(αxc Sy2 + βy c Sx2 ) [2(αxc + βy c ) − bαβxc y c (Sx + Sy )]−3b2 αβxc y c (Sx + Sy )2 .

The curve of J3 = a1 a2 a3 − (a0 a23 + a21 a4 ) = 0 is the partition line between
the stable and unstable regions. Substituting
a4 =a1 a2 a3 − a0 a23
a21

into the characteristic polynomial and factoring it yield(a3 + a1 λ2 )(a1 a2 − a0 a3 + a21 λ + a0 a1 λ2 ) = 0
Two of the characteristic roots are purely imaginary,ra3
λ1,2 = ± − = ±iξ,a1
and the other roots are the solutions of the quadratic equation
a0 a1 λ2 + a21 λ + a1 a2 − a0 a3 = 0,
which have negative real parts, since all coeﬃcients are positive. The appearanceof Hopf bifurcation as the value of Sx crosses the partition line can be examined
similarly to the previous case.
Case 3. Sx > 0 with nx = 1 and Sy > 0 with ny = 1.
In the third special case we assume that both firms have information lags
about their outputs and the weighting function has a bell-shaped curve peakedaround t − s = Si for i = x, y. Substituting these values into the characteristic
equation and arranging terms indicate that the eigenvalues are the roots of thefollowing sixth degree polynomial,ϕ(λ) = a0 λ6 + a1 λ5 + a2 λ4 + a3 λ3 + a4 λ2 + a5 λ + a6 ,where the coeﬃcients are defined as
a0 = (Sx Sy )2 > 0,
a1 = 2Sx Sy (Sx + Sy ) > 0,
a2 = Sx2 + 4Sx Sy + Sy2 − b2 αβxc y c Sx2 Sy2 ,a3 = 2(Sx + Sy ) + 2b(αxc Sy2 + βy c Sx2 ) − 2b2 αβxc y c Sx Sy (Sx + Sy ),a4 = 1 + 4b(αxc Sy + βy c Sx ) − b2 αβxc y c (Sx2 + 4Sx Sy + Sy2 ),a5 = 2b(αxc + βy c ) − 2b2 αβxc y c (Sx + Sy ),a6 = 3b2 αβxc y c > 0.
16

The Routh-Hurwitz determinants are defined as follows: J2 = a1 a2 − a0 a3 ,¯¯¯¯ a1 a0 0 0¯¯¯ a1 a0 0 0 ¯¯¯¯ a1 a0 0 ¯¯¯ a3 a2 a1 a0¯¯¯ a3 a2 a1 a0 ¯
¯
¯
¯
¯
¯
a
a
a
and J5 = ¯¯ a5 a4 a3 a2
J3 = ¯ 32
1 ¯ , J4 = ¯¯¯ a5 a4 a3 ¯¯ a5 a4 a3 a2 ¯
¯ 0 a6 a5 a4¯ 0 a6 a5 a4 ¯¯
¯ 0 0 0 a6

¯
0 ¯¯
0 ¯¯
a1 ¯¯ .
a3 ¯¯
a7 ¯It is very hard to analyze the signs of these determinants analytically, therefore we perform numerical simulations. We take a = 10, b = 1, α = β = 1 andcx = cy = 1 and numerically confirm that the J5 = 0 curve is the partition line
in Figure 5 where the curves, a2 = 0, J3 = 0 and J4 = 0 are located outside theregion shown in the figure. In the shaded region below the J5 = 0 curve, theRouth-Hurwitz stability conditions are satisfied.

Figure 5. Tthe partition line and stable region
We also numerically confirm that with the selection of Sx = 0.4 and Sy =0.2 the dynamic system (17) generates complex dynamics. It is illustrated in
Figure 6 where the time trajectory is plotted in the (ln x, ln y) space. It is wellknown that the weighting function (9) converges to the Dirac delta function
when nx and ny tend to infinity. We also showed earlier that the characteristic
polynomial of continuously distributed delay models also converge to that of
the fixed delay, so in the limiting case, the model with continuously distributed
time lag becomes identical with the fixed time delay model. Comparing Figure
3 with Figure 6 reveals the similarity between dynamics generated by the modelwith continuously distributed time lag and dynamics by the fixed time delay

17

model even when nx = ny = 1.

Figure 6. Trajectory in the (ln x, ln y) space

4.2

Eﬀects of the information lag about the rival’s output

In this section we assume again that there are no information lags about the own
outputs of the firms, similarly to the model introduced in Section 3.1, however
we assume the nonlinear adjustment process (15). In this case the characteristic
equation (11) has the formµλTx
(λ+2bαx )(λ+2bβy ) 1 +
q̄x
c

c

¶mx +1 µ¶my +1λTy−b2 αβxc y c = 0. (18)1+q̄yThis is the same equation as (12) when α and β are replace by αxc and βy c ,respectively. In Section 3.1 we have proved that all roots of equation (12)have negative real parts, which also holds for equation (18) showing that thestationary state is always locally asymptotically stable.

5

Conclusion

Dynamic duopolies were examined with linear price and cost functions. We
assumed first constant speeds of adjustment in a gradient adjustment process.
It is well-known that the stationary state is always globally asymptotically stable. If the firms have fixed or continuously distributed time delays only on the
outputs of the rivals, then stability is preserved. The stability might be lost if
the firms have delays in their own outputs.
Similar results were shown in the case when the firms adjust their growth
rates of the outputs proportionally to the changes in their profits. The resulting
dynamic systems are nonlinear, and we have shown that similar results hold18

about the local asymptotical stability of the steady state. In the case of stability
loss Hopf bifurcation occurs giving the possibility of the birth of limit cycles
around the stationary state.

19

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