Dynamics in Linear Cournot Duopolies with
Two Time Delays∗
Akio Matsumoto†Chuo University

Ferenc Szidarovszky‡
University of Arizona

Hiroyuki Yoshida§
Nihon University

Abstract
Linear differential duopolies are constructed with continuous time scales,
constant coefficients and two types of information delays: fixed and continuously distributed time delays. System dynamics are considered withdelays in the diagonal terms. By analyzing the associated characteristic
equation, it is found that the stability is lost when the lengths of delays
cross some critical value. Then it is shown that the destabilizing effectcaused by the fixed delays is stronger than the destabilizing effect of the
distributed delay having exponentially-declining weighting function. It
is further demonstrated that the strength of the destabilizing effect isreversed if the distributed delay has a bell-shaped weighting function.

Keywords: Cournot competition, fixed time delay, continuously distributed time delay, two delays, stability switches∗ The earlier verion of this paper was presented at MDEF 2010 (Modelli Dinamici in Economic e Finanza) held in Urbino, Itlay, September 23-25, 2010. The authors are grateful tovaluable comments by Tony Hey, Laura Gardini, Gian-Italo Bischi, Volker Böhm and Barkley
Rosser. The authors greatly acknowledge financial supports from the Japan Society for the
Promotion of Science (Grant-in-Aid for Scientific Research C, 21530172), Chuo University(Joint Research Grant 0981) and the Japan Economic Research Foundation. 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.jp1

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 generalizations 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. (2010) whichcontains 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 this paper continuous-time gradient dynamics will beconsidered with constant coefficients. Some applications of gradient dynamics
to economic models are found in Bischi et al. (2010).
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) construct two linear continuous-time dynamics
oligopoly models with partial adjustment towards the best response and consider
the effects caused by time delays on local stability. They show clear-cut results.
In their Model I in which each firm has delayed information of its own output
and its competitors’ outputs, local stability can be lost when the lengths of thedelays are large enough. On the other hand, in their Model II in which eachfirm has delayed information only in the competitors’ outputs, stability is preserved regardless of the length of the delays under the plausible assumption thatthe coefficient matrix of the dynamic system without time delays is diagonallydominant. They, however, do not investigate the case in which each firm hasdelayed information on its own output and can use instantaneous informationon the competitors’ outputs.
This paper has three purposes. The first purpose is to complement the
study of Howroyd and Russel (1984) by examining how the fixed time delays
affect local dynamics in this missing case. Howroyd and Russel (1984) assume
fixed time delays. There are, however, many economic situations in which thedelays are uncertain or the firms are reacting to averaged past information. In
such situations, continuously distributed time delays are appropriate. Thus thesecond purpose is to examine whether or not the results obtained under fixed
time delays still hold in a dynamic oligopoly model with continuously distributed
time delays. The last purpose is to compare the destabilizing effects caused by
fixed and distributed time delays and to show that the relative strength between
2the two effects depends on the value of the shape parameter of the weighting
function under distributed delay.
The paper is organized as follows. Section 2 constructs a basic duopoly
model with linear price and cost functions. Section 3 assumes constant speed ofadjustment and introduces fixed time delays into the basic model. Section 4 discusses the model with continuously distributed time delay. Section 5 concludes
the paper.

2

The Basic ModelIn this section, dynamics in a classical linear oligopoly model is briefly reviewed
and its global stability is confirmed. Consider an industry of N firms producing
a homogeneous good. Let k = 1, 2, ..., N denote the firms and let xk be theoutput quantity produced by firm k. The price function is assumed to be linear,
p=a−b

N


xℓ with a > 0 and b > 0.

ℓ=1

Production cost is also assumed to be linear and the marginal cost of firm k isdenoted by ck . The profit function of firm k is defined byNπk = a − bxℓ xk − cx xkℓ=1

Firm k determines its output to maximize its profit with respect to xk . Assuming interior optimal solutions of the profit maximizing problems and thensolving the first-order condition for the output yield the best reply for firm k,

Rk (x−k ) =a − ck − b

N


xℓ

ℓ=k

2b

with
x−k = (x1 , x2 , ...xk−1 , xk+1 , ..., xN ).
A Cournot equilibrium state is a vector (xc1 , xc2 , ..., xcN ) that satisfies xck =Rk (xc−k ) for k = 1, 2, ..., N. Thus a Cournot equilibrium output of firm k
is
N

cℓ
a − Ncx +
xck =

ℓ=k

(N + 1)b.

In this paper we assume that the firms continuously adjust their outputs
proportionally to the change in their profits. This gradient dynamics is modeledby an N-dimensional system of ordinary differential equations of the form,ẋk = αk

∂πk
for k = 1, 2, ..., N ,
∂xk

(1)

where αk is a positive adjustment coefficient of firm k, and the dot over a variablemeans its time derivative, e.g., ẋk = dxk /dt. Substituting the explicit expression3

of the marginal profit of firm k into (1), we obtain the continuous-time oligopoly
model with gradient dynamicsNẋk (t) = αk a − ck − bxℓ (t) − 2bxk (t) for k = 1, 2, ..., N .
(2)
ℓ=k

Using the best reply functions, system (2) can be rewritten asẋk (t) = ᾱk (Rk (x−k (t)) − xk (t)) for k = 1, 2, ..., N .

(3)where ᾱk = 2bαk . In system (3), each firm adaptively adjusts its output in sucha 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 transformationfrom (2) to (3) or vice versa implies that for the firms, the gradient adjustmentof the output is the same as the adaptive adjustment toward best reply. To
establish local stability of the Cournot equilibrium, it suffices to show that all
eigenvalues of the coefficient matrix of either (2) or (3) have negative real parts.
It is shown in Bischi et al. (2010, Theorem 2.2) that system (3) is always locallyasymptotically stable. This implies that system (2) is also locally asymptoticallystable. Since local stability leads to global stability in linear models, we obtainthe following well-known result:
Theorem 1 The continuous-time gradient dynamic model (2) is globally asymptotically stable.
In the following we introduce time delays into system (2) and consider how
time delays affects local stability of the Cournot equilibrium. We examine twodifferent types of time delays: fixed or discrete time delay in Section 4 andcontinuously distributed time delay in Section 5. We abbreviate the first delay
to a discrete-delay and the second to a continuous-delay henceforth.

3

Linear Duopolies with Two Discrete Delays

Howroyd and Russel (1984) obtain the delay oligopoly system by introducing
discrete-delays into (3),ẋk (t) = ᾱk (Rk (x−k (t − Tk )) − xk (t − Sk )) for k = 1, 2, ..., N ,where firm k experiences a time delay Tk in obtaining information about thecompetitors’ outputs and a time delay Sk in implementing information about
its own output. The Cournot equilibrium is shown to be stable in their ModelI with Tk = Sk = rk > 0 if 2ᾱk rk ≤ 1 and (N − 1)b < 1 hold. It is also shown
to be always stable regardless of the lengths of delays in their Model II with
Tk > 0 and Sk = 0.1 In the following, we consider the opposite case with Sk > 0
and Tk = 0 in the gradient dynamic model (2). Since it is, however, difficult to
examine the general N -firm oligopoly model, we confine our analysis to delay
duopoly dynamics (i.e., N = 2).
1 More general result, asymptotic stability for linear differential system with general delays

Tkj ≥ 0 for 1 ≤ k = j ≤ N and Skk = 0 for k = 1, 2, ..., N, is shown in Hofbauer and So(2000).

4

For the sake of notational simplicity, we use the followings: x = x1 , y =x2 , α = α1 , β = α2 , cx = c1 and cy = c2 . Then a duopoly version of thegradient dynamic model is presented byẋ(t) = α(a − cx − 2bx(t − Sx ) − by(t − Tx )),(4)ẏ(t) = β(a − cy − bx(t − Ty ) − 2by(t − Sy )).
A steady state (i.e., Cournot outputs) of the delay model (4) is obtained bysubstituting 2 for N in xck ,xc =a − 2cx + cy
a − 2cy + cx
and y c =,3b3bwhich are independent of the length of delays. Since the discrete-delays in theoff-diagonal terms are harmless to stability, we assume the following:
Assumption 1 Tx = Ty = 0.
We start from the local stability analysis. For this purpose we consider thecorresponding homogenous system of (4):ẋδ (t) = α(−2bxδ (t − Sx ) − byδ (t)),
ẏδ (t) = β(−bxδ (t) − 2byδ (t − Sy )).
where xδ (t) = x(t)−xc and yδ (t) = y(t)−y c are deviations from the corresponding equilibrium values. The characteristic equation of the linearized system canbe obtained by looking for the solutions in exponential forms,xδ (t) = eλt u and yδ (t) = eλt v
and substituting them into the homogeneous equations. The resultant system
can be written in the matrix form:

  

λ + 2αbe−λSx
0
αb
u

= 

0
v
βb
λ + 2βbe−λSy
A nontrivial solution exists if and only if the determinant of the coefficientmatrix is zero, which provides a mixed exponential-polynomial equation for λ:λ + 2αbe−λSx

λ + 2βbe−λSy − b2 αβ = 0.

(5)Expanding (5), we obtain the characteristic equation of the form
λ2 − αβb2 + 2αbλe−λSx + 2βbλe−λSy + 4αβb2 e−λ(Sx +Sy ) = 0.

(6)

Due to Theorem 1, the duopoly system (4) is locally asymptotically stablein the absence of information delays. Although we assume away information
delays in the competitors’ outputs and reduce the N-firm model to the duopoly
model, the dynamic analysis of system (4) is still complicated in the case ofmultiple distinctive delays, Sx > 0, Sy > 0 and Sx = Sy . Following Li and Wei(2009), we take the two-step procedure. At the first step, setting Sx = 0, we
5reduce (6) to the equation with single discrete-delay Sy . We first find a stableinterval of Sy in which all roots of the characteristic equation (6) have negative
real parts. Then we proceed to the question of stability switching. The stability
analysis concerns with whether all roots lie in the left half of the complex plain.
The analysis of stability switching concerns with whether the roots cross theimaginary axis when the delay changes. At the second step, selecting a value ofSy from the stable interval, we treat Sy as a parameter and find a stable intervalof Sx , repeating the same procedure taken at the first step. The stable intervalof Sx depends on the value of Sy . Moving the value of Sy from one extreme of its
stable interval to the other extreme generates the Sy -dependent corresponding
interval of Sx . Putting these intervals together constructs the region of Sx and
Sy in which the Cournot equilibrium is stable.
First Step : Sx = 0 and Sy > 0.
The condition Sx = 0 reduces the characteristic equation (6) to the form,
λ2 + 2αbλ − αβb2 + 2βb(λ + 2αb)e−λSy = 0.

(7)

It can be checked that λ = 0 is not a solution of the above equation. Let λ = iωwith ω > 0 be a solution. Substituting it into (7) and separating the real andimarinary parts, we have2βb(2αb cos ωSy + ω sin ωSy ) = ω2 + αβb2
(8)2βb(ω cos ωSy − 2αb sin ωSy ) = −2αbω.
The sum of the squares of these two equations yields the fourth-order equationin ω(9)
ω4 − 2b2 (2β 2 − 2α2 − αβ)ω 2 − 15α2 β 2 b4 = 0
implying that

ω 2± = b2 (2β 2 − 2α2 − αβ ±√D)where D is the discriminant and has the form
(2β 2 − 2α2 − αβ)2 + 15α2 β 2 b4 > 0.
Clearly ω 2+ > 0 and ω2− < 0. Substituting the positive root ω + into (8) andsolving the resultant equations we havecos ω + Sy =andsin ω + Sy =α2 b2
4α2 b2 + ω2+

ω + (ω 2+ + αβb2 + 4α2 b2 ).
2βb(4α2 b2 + ω2+ )Hence there is a unique θ = ω + Sy ∈ (0, 2π] makes both equations hold. Solving
either of the last two equations for Sy yields the threshold value of time delay
Sy , given Sx = 0,1
α2 b2
SyD0 =cos−1ω+4α2 b2 + ω 2+
6

or
SyD0 =

ω + (ω 2+ + αβb2 + 4α2 b2 )
1
sin−1
ω+
2βb(4α2 b2 + ω2+ )where the values of the right hand sides of these expressions are the same.
To verify stability switching, we need to determine the sign of the derivativeof Re[λ(Sy )] at the point where λ(Sy ) is purely imaginary. By differentiating(7) with respect to Sy , we have2 2(λ + αv + βbe−λSy ) − βb(λ + 2αb)e−λSy Sy dλ= 2βb(λ + 2αb)λe−λSy .
dSy

For convenience we check the sign of (dλ/dSy )−1 that is written as


dλ
dSy

−1

=

(λ + αb)eλSy + βb Sy−
βbλ(λ + 2αb)λwhere, from (7), we have2βb(λ + 2αb).
eλSy = − 2
λ + 2αbλ − αβb2
Therefored(Re λ)sign
dSy λ=iω+dλ= sign Re
dSy

−1 

λ=iω+

1
2(λ + αb)+= sign −22λ(λ + 2αbλ − αβb ) λ(λ + 2αb) λ=iω+2(ω 2+ + αβb2 + 2α2 b2 )
1
− 2 2
2
2
2
2
(ω + + αβb ) + (2αbω+ )4α b + ω 2+
= sign 2(ω 2+ + αβb2 ) + 4b2 (α2 − β 2 ) √ = sign 2b2 D > 0.
= signThis result implies that the crossing of the imaginary axis is from the left to theright as Sy increases and thus leads to the loss of stability. It can be noticed
that the firms are symmetric in the adjustment coefficients and discrete-delays.
We can get the threshold value of Sx by interchanging α and β. Concerning
stability of the Cournot equilibrium, the following conclusion holds at the first
step:
Theorem 2 The discrete-delay duopoly model with Si > 0 and Sj = 0 (i, j =x, y, i = j) is locally asymptotically stable for Si < SiD0 and unstable for Si >SiD0 where the critical level of discrete-delay SiD0 is defined by


2 2
b
b
1
cos−1
SiD0 =
ω i+
4b2 b2 + ω 2i+
with



ω i+ = b 2a2 − 2b2 − αβ + Di
7

andDi = (2a2 − 2b2 − αβ)2 + 15α2 β 2 b4

where

a = α and b = β if i = x and a = β and b = α if i = y.

Second Step : Sx > 0 and Sy > 0.
Suppose that the characteristic equation (6) has a value of Sy selected fromthe stability interval [0, SyD0 ) obtained at the first step. By the same token, let
λ = iw, w > 0 be a solution of the characteristic equation. Substituting it into(6) we have the real and imaginary parts in the following forms:
2αbw sin wSx + 4αβb2 cos w(Sx + Sy ) = w2 + αβb2 − 2βbw sin wSy ,(10)2αbw cos wSx − 4αβb2 sin w(Sx + Sy ) = −2βbw cos wSy .
Hence the sum of the squares of the two equations in (10) is arranged to be
4α2 b2 w2 + 16α2 β 2 b4 − 16α2 βb3 w sin wSy= (w2 + αβb2 )2 + 4β 2 b2 w2 − 4βb(w2 + αβb2 )w sin wSy(11)which is written as a fourth-order equation in w,
w4 + 2b2 (2β 2 − 2α2 + αβ)w2 − 15α2 β 2 b4 + 4βbw(4α2 b2 − w2 − αβb2 ) sin wSy = 0.
It is not easy to solve this equation analytically for w. However, under a specificcondition such as α = β = k, the equation is reduced to(w2 − 3k2 b2 )(w2 + 5k2 b2 − 4kbw sin wSy ) = 0.
Since the second factor √is always positive, a positive solution of w is uniquelydetermined to be w+ = 3kb which is independent of the specific value of Sy .
Using the addition theorem for trigonometric functions, we solve (10) to
obtainsin wSx =2αb(w2 + αβb2 ) + 8αβ 2 b3 w − 4αβb2 (αβb2 + 2w2 ) sin wSy,
4α2 b2 (w2 + 4β 2 b2 − 4βb sin wSy )andcos wSx =β 2 b2 cos wSy.
w2 + 4β 2 b2 − 4βb sin wSy

Again, we need to determine the sign of the derivative of Re[λ(Sx )] at the
point where λ(Sx ) is purely imaginary. From (6), we have
−1(2λ + 2βb(1 − λSy )e−λSy )eλSx
2αb − 4αβb2 e−λSy
Sx
dλ
=
+
−
−λS
2
y)
dSx
λ(2αbλ + 4αβb eλ(2αbλ + 4αβb2 e−λSy )λwhere from (6)2αbλ + 4αβb2 e−λSy
eλSx = − 2.
λ − αβb2 + 2βbe−λSy8Therefored(Re λ)dSx λ=iw+
 
−1 
dλ
= sign Re
dSx
sign

λ=iw+

2λ + 2βb(1 − λSy )e−λSy
2αb − 4αβb2 Sy e−λSy= sign −
+
2
−λS
2
y)
λ(2αbλ + 4αβb2 e−λSy ) λ=iw+
λ(λ − αβb + 2βbλe

22+ αβb2 ) sin w+ Sy + 2βbw+ (αβb2 + w+
)Sy cos w+ Sy−4β 2 b2 w+ + 2αb(2w+= sign −
2 2 2
2
2
2
2
w+ (4β b w+ + (αβb + w+ ) − 4βb(αβb + w+ )w+ sin w+ Sy )

8α2 βb3 sin w+ Sy + 8α2 βb3 w+ Sy cos w+ Sy − 4α2 b2 w++
2 + 16α2 β 2 b4 − 8α2 βb3 w sin w S )w+ (4α2 b2 w+
+
+ y

2
4
2
2
2
2
= sign 8αβb (α − β ) + 2w+ (αβb + w+ ) + 2βb(4α2 b3 − αβb2 − 3w+) sin w+ Sy

2 3
2
2
+ 2βb(4α b − αβb − w+ )w+ Sy cos w+ Sy

where we use (11) in the last step. Under the specific condition α = β = k, theexpression in the last brackets is simplified as√22) − 4kbw+
sin w+ Sy > (8 3 − 12)k2 b3 > 0.
2w+ (k2 b2 + w+√
The first inequality is due to |sin w+ Sy | < 1 and w+ = 3kb. Therefore we
obtaind(Re λ) >0dSx λ=iw+

implying that all the roots cross the imaginary axis from left to right as Sx
increases when Sy is chosen from its stability interval. The result is summarized
in the following theorem:
Theorem 3 Given α = β = k, the discrete-delay duopoly model with Sx > 0
and Sy ∈ (0, SyD0 ] is locally asymptotically stable for Sx < S̄x and unstable for
Sx > S̄x where S̄x = S̄x (Sy ) and S̄x (Sy ) is defined by
β 2 b2 cos w+ Sy
1−1S̄x (Sy ) =cos
2 + 4β 2 b2 − 4βb sin w S
w+
w+
+ y
with
w+ =

√
3kb.

Notice that S̄x (Sy ) converges to the threshold value SxD0 obtained in Theorem 2, when Sy approaches 0. The locus of S̄x = S̄x (Sy ) for Sy ∈ (0, SyD0 ] isthe partition curve dividing the discrete-delays (Sx , Sy ) space into stable andunstable regions and is depicted as a red concave curve in Figure 1B.

4

Linear Duopolies with Two Continuous-Delays

In real economic situations, the delays are usually uncertain and can be considered to be fixed only under special circumstances. Therefore, we will re-model
9

time delays in a continuously distributed manner and compare the discretedelay effects with the continuous-delay effects. As in the discrete-delay model,
we examine the case where the firms have time delays only in implementing information about their own outputs.2 State variables x(t) and y(t) in the diagonalterms are replaced by certain averages of past values. The continuous-delay
dynamic system is frequently modeled with Volterra type integro-differential
equationsẋ(t) = α(a − cx − 2bxe (t) − by(t)),(12)ẏ(t) = β(a − cy − bx(t) − 2by e (t)),
with expectations on its own outputs, txe (t) =φ(t − s, Sx , nx )x(s)ds0andy e (t) = t0φ(t − s, Sy , ny )y(s)ds.

The weighting function φ is assumed to have the form
 1 t−s

e− Γ


 Γ
φ(t − s, Γ, ℓ) = ℓ+1ℓ(t−s)1 ℓ(t − s)ℓ e− Γℓ! Γ

if ℓ = 0,(13)
if ℓ ≥ 1.

Here we assume that Γ > 0 and ℓ is a nonnegative integer. For ℓ = 0, weights are
exponentially declining with most weight given to the most current data. For ℓ >0, the weighting function has a bell-shaped profile indicating that zero weightis given to the most recent data, rising to maximum at s = t − Γ and declining
thereafter. As ℓ increases, the weighting function becomes more peaked aroundΓ and as ℓ goes to infinity, the weighting function converges to the Dirac delta
function.3
Substituting (13) into the above expectation formations, and the resulting
expressions into (12) yield a system of integro-differential equations. A steadystate of system (12) is identical with the Cournot equilibrium, xc and yc . In orderto analyze the dynamic behavior of the system, we consider the corresponding
homogeneous system. Letting xδ and yδ , as before, denote the deviations of x
and y from their Cournot output levels, xc and y c , the homogeneous system canbe formulated as follows: tẋδ = α −2b φ(t − s, Sx , nx )xδ (s)ds − by(t) ,0 tẏδ = β −bx(t) − 2b φ(t − s, Sy , ny )yδ (s)ds .

(14)

0

We seek the solutions in the exponential formsxδ (t) = eλt u and yδ (t) = eλt v,
2 It is possible to show that time delays in the competitor’s output are harmless to stability

even in the continuous-delay model. See Matsumoto and Szidarovszky (2009)
3 See Bischi et al. (2010) for more properties of the weighting function, φ(t − s, Γ, ℓ).

10and substituting these solutions into equation (14) and arranging terms yield t−λ(t−s)λ + 2αb φ(t − s, Sx , nx )e
ds u + αbv = 0,
0



 t
−λ(t−s)ds v = 0.
βbu + λ + 2βb φ(t − s, Sy , ny )e

(15)

0

Introducing a new variable z = t − s, we can simplify the integral terms bynoticing that t t−λ(t−s)φ(t − s, Γ, ℓ)eds =φ(z, Γ, ℓ)e−λz dz.
00Allowing t → ∞, we have
lim

t→∞

−(ℓ+1)λΓφ(z, Γ, ℓ)e−λz dz = 1 +
q
0

 t

with
q=


 1 if ℓ = 0,ℓ

if ℓ ≥ 1.

Then equations (15) can be written in the matrix form


  
Ax (λ) Bx (λ)
u0=By (λ) Ay (λ)
v
0
where

and

 

n +1
λSx x
+ 2αb ,Ax (λ) = λ 1 +
qx
 

n +1
λSy y
+ 2βb ,Ay (λ) = λ 1 +
qy

n +1
λSx x
Bx (λ) = αb 1 +
qx
n +1

λSy y
By (λ) = βb 1 +.
qy

A non-trivial solution exists if and only if
Ax (λ)Ay (λ) − Bx (λ)By (λ) = 0(16)which is called the characteristic polynomial of system (14). Since it is difficult
to check whether the real parts of the roots of (16) are negative or positive, wewill specialize the from of the density function to obtain some analytical results.
We have already assumed that both firms have information delays about
their own outputs, Sx > 0, Sy > 0 and Sx = Sy . In addition to this, we further
assume that the firms have exponentially declining weighting functions, that is,11nx = ny = 0. Then equation (16) under these assumptions becomes a 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  0,
a3 = 2b(α + β) − b2 αβ(Sx + Sy )  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  and J3 =  a3 a2 a1  .
J2 = 

a3 a2  0 a4 a3 

Notice that
andJ2 = Sx + Sy + 2b(αSy2 + βSx2 ) > 0
J3 = a3 J2 − a21 a4 .

Since J2 , 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. It is thus sufficient forour purpose to assume that J3 > 0, which implies that all roots have negative
real parts.
Theorem 4 If Sx > 0 with nx = 0 and Sy > 0 with ny = 0, then thecontinuous-delay system (12) is globally asymptotically stable for (Sx , Sy ) below the partition curve, J3 = 0, and unstable for (Sx , Sy ) above the curve whereJ3 = b Sx + Sy + 2b(αSy2 + βSx2 ) [2(α + β) − bαβ(Sx + Sy )] − 3bαβ(Sx + Sy )2 .

The graphical representation of Theorem 4 is given in Figure 1A where
the downward sloping curve is the locus of J3 = 0 with α = β = b = 1. In
the horizontally-shaded region under the locus, we have J3 < 0 and thus the
Cournot equilibrium is asymptotically stable. Figure 1B is an enlargement of
the square region surrounded by the two loci, Sx = SxD0 and Sy = SyD0 , in thelower-left part of Figure 1A. The downward-sloping red curve is the partition
curve under the two discrete delays being within the horizontally-shaded regionof Figure 1A. Therefore the stability region with the discrete-delay is smaller
than the stability region with the continuous delay with nx = ny = 0.
Furthermore, if we solve J3 = 0, taking Si > 0 and Sj = 0, (i, j = x, y, i = j),then we obtain the threshold value of the continuous-delay of firm i in the casewhere firm i has continuous-delay information on its own output while firm j
has instantaneous information about its own output,
c + α2 + αβ + β 2
C0
Si =
αβb
12

with
c = β if i = x and c = α if i = y.
The continuous-delay system with Si > 0 and Sj = 0 is asymptotically stable
if 0 < Si < SiC0 and unstable otherwise. This result is also visualized in Figure
1A where the outer dotted horizontal or vertical line is the locus of Si = SiC0 .
Under the horizontal line or in the left of the vertical line, the Cournot outputs
are asymptotically stable when only one firm experiences a continuous-delay
with exponentially declining weighting function.
It can be also observed in Figure 1A that the stability region with two
continuous-delays is smaller than the stability region with a single continuoustime delay. The partition line under the single discrete-delay, Si = SiD0 , is also
depicted in the shaded region as the inner horizontal or vertical dotted line. Thispartition curve with two discrete-delays is depicted as the outermost concaveshaped curve in Figure 1B where the stability region with two discrete-delays issmaller than the stability region with a single discrete-delay. Locations of thesedotted lines indicate that the destabilizing effect of the single discrete-delay isstronger than the destabilizing effect of the single continuous-delay in the sensethat the stability region with the discrete-delay is smaller than the stability
region with the continuous delay.
It is possible to derive an analytic form of the partition curve with larger
values of nx and ny . It becomes, however, clumsy and much more complicated.
Therefore we numerically check the shapes of the partition curves. Taking
α = β = b = 1 again and repeating the same procedure with increasing valuesof nx = ny = 1, 2, 3, 4, we obtain the four partition curves illustrated in Figure
1B in which Pi means the partition curve when nx = ny = i. It can be seenthat the continuous-delay partition curve moves outward as the value of nx =ny increases. In other word, the stability region under the continuous-delays
expands and get closer to the stability region under the discrete-delays.
There is an interesting relation between the characteristic polynomials (6)and (16) of the systems with discrete- and continuous-delays. Assume that nx
and ny converge to infinity. Since qi = ni for i = x, y, we have the limits
Ax (λ) → λeλSx + 2αb ,Ay (λ) → λeλSy + 2βb ,Bx (λ) → αbeλSx
andBy (λ) → βbeλSy .

Multiplying both sides of the characteristic equation (16) by e−λSx e−λSy andarranging terms, we get the characteristic equation (5),λ + 2αbe−λSx

λ + 2βbe−λSy − αβb2 = 0.

Therefore in the limiting case, the characteristic equation of the continuousdelay model converges to the characteristic equation of the discrete-delay model.
The former model can approximate dynamics generated by the latter model if

13the value of the shape parameters (i.e., nx and ny ) of the weighting functions
is large enough.

(A) nk = 0 for k = x, y(B) nk = 1, 2, 3, 4 for k = x, y

Figure 1. Partition lines of the discrete- and continuous-delay models.
In short, we find that increasing the shape parameters of the weighting
function has a stabilizing effect in the sense that the stability region in theparameter space enlarges as the values of nx and ny increase.

5

Concluding Remarks

It is well-known that a continuous-time linear oligopoly is globally asymptotically stable. In this paper, we introduce two types of a time delay, a discretedelay and continuous-delay, and consider the destabilizing effects caused by thetime delay in the duopoly framework. We draw attention to the case where
each firm has delay information only on its own output and obtain the following
results.
(1) The analytical form of the partition curve dividing the (Sx , Sy ) space into
the stability and instability regions is explicitly derived as summarized in
Theorems 2, 3 and 4.
(2) The destabilizing effect caused by the discrete-delay is stronger than thedestabilizing effect by the continuous-delay having the exponentially declining weighting function (i.e., nx = ny = 0) in the sense that the stability
region with the former is smaller than the stability region with the latteras depicted in Figure 1A.
(3) The strength of the destabilizing effects is reversed if the continuous-delay
has the bell-shaped weighting function (i.e., nx = ny > 0) as shown in
Figure 1B.
(4) The stability region with continuous-delay expands as the shape parameterof the weighting function increases and converges to the stability regionof discrete-delay when the shape parameter tends to infinity.

14

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