Dynamics in Delay Monopoly∗
Akio Matsumoto†Chuo University

Ferenc Szidarovszky‡
University of Arizona

Abstract
The purpose of this paper is to study the dynamics of a delay differential monopoly with the gradient method. The paper is divided into
two parts. First we assume constant speed of adjustment and investigate stability switches of a stationary state by analyzing the corresponding characteristic equation in cases of single fixed time delay, two fixed
time delays and a continuously distributed time delay. Second, we adopt
output-dependent speed of adjustment and demonstrate the birth of Hopf
bifurcation with respect to the time delay when a stationary state is locallyunstable.
Keywords: delay monopoly, single and multiple fixed time delay,
continuously distributed time delay, Hopf bifurcation

JEL Classification: D42∗ The 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, Higashi-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

11Introduction

In recent year an increasing attention has been given to the periodic and aperiodic behavior of systems that can be described by delay diﬀerential equation (seefor example, Bélair and Mackey,1989 and Invernizzi and Medio, 1991). With
the infinite dimensionality created by delays, even a single first-order equationis transformed into an equation with a suﬃcient number of degrees of freedom
to permit the occurrence of chaotic phenomena (e.g., see an der Heiden and
Mackey (1982)). This finding indicates that delay models of a dynamic economy may explain various complex dynamic behavior of the economic variables.
It also indicates that in modelling firm’s dynamic behavior, it is necessary andessential to consider time delays inherent in phenomena like information andimplementation delays. In spite of this, it is usually assumed that the economic
agents have instantaneous information about their own behavior and also on
the competition’s behavior mainly due to mathematical convenience.
In constructing dynamic economic models, the most common processes are
based on either the gradients of the profit functions or the best replies of theagents. Time delays can be modeled in two diﬀerent ways: fixed time delay
and continuously distributed time delays. Chiarella and Khomin (1996) and
Chiarella and Szidarovszky (2001) examine delay diﬀerential oligopolies withbest replies by using continuously distributed time delays. Howroyd and Russel(1984) detect the stability conditions of delay output adjustment processes ina general N -firm oligopoly with fixed time delay. In the case of the gradientmethod, we mention the works of Bischi and Naimuzada (1999) and Bischi and
Lamantia (2002) for discrete oligopoly dynamics without time delays and theworks of Puu (2003) and Ahmad et al. (2000) for discrete monopoly dynamics
with time delays. Recently, Matsumoto et al. (2009) introduce the gradientmethod into delay diﬀerential Cournot models and investigate their dynamics
under both fixed and continuously distributed time delays. Although it has
been shown by Ahmad et al, (2000) that delay increases stability and by Puu(2000) that chaos can emerge in delay discrete monopoly, dynamics of a delay
diﬀerential monopoly with the gradient method has not yet been revealed in
the existing literature. The main purpose of this study is to demonstrate thepossibility of the birth of limit cycles in such a delay diﬀerential monopoly withthe gradient method.
The paper is organized as follows. Section 2 constructs a basic monopoly
model with linear price and cost functions and then introduces time delay into
the basic model provided that the speed of adjustment is constant. Section 3
discusses a nonlinear extension of the basic model by adopting the growth rateadjustment process. Section 4 concludes the paper.

2

Delay Linear Monopoly

Dynamics in a monopoly model with delays is considered. The model is simple
and standard but suﬃces to bring out cyclic fluctuations in output production.

2

Let q be the output produced by a monopoly firm with a unit production cost,c. The price function is assumed to be linear
f (q) = a − bq, a > 0 and b > 0.
Then the profit of the monopoly firm is given by
π = (a − bq)q − cq.
The optimal output that maximizes the profit is obtained from the first-ordercondition for an interior solution,
qM =

a−c
2b

where a > c is assumed for the positive equilibrium output. In the text bookmodel, it is usually implicitly assumed that the monopoly firm has full information on the entire demand curve. As a result, such a monopoly firm can jump to
the profit maximizing output. The more recent literature of dynamic processes
raises doubts on the assumption of full rationality and adopts bounded rationality in a sense that a monopoly firm has only limited or partial informationon the demand curve. Under such a circumstance, a more racialistic approachis to formulate the dynamic process with the gradient method:q̇ = α(q)

dπ
dq

(1)

where α(q) is a positive function that gives the adjustment of the firm. According to (1), the firm changes its quantities at a rate proportional to themarginal profit. In constructing best response dynamics global information isrequired about the profit functions, however in applying gradient dynamics only
local information is needed. We start with a simple case and assume a constantadjustment coeﬃcient in (1):Assumption 1. α(q) = α > 0.
The gradient dynamics with a fixed time lag τ > 0 under Assumption 1 ispresented byq̇(t) = α [a − c − 2bq(t − τ )] .
(2)

Introducing the new variable x(t) = q(t) − q M reduces (2) toẋ(t) = −γx(t − τ ) with γ = 2bα > 0,(3)which is a first-order delay linear equation with a trivial solution x(t) = 0 (orq(t) = q M ) for all t ≥ 0. If there is no time lag, equation (3) generates a solution,x0 e−γt , that converges to the trivial solution as t approaches infinity. Even in
the case of the delay equation, substituting the exponential solution x(t) =x0 eλt into (3) and then arranging terms yields its corresponding characteristic
equationλ + γe−λτ = 0.
(4)
3

By continuity, it can be supposed that (3) is stable for a small value of τ . As thelength of the delay changes, the stability of the trivial solution may also change.
Such phenomena is referred to as a stability switch. In order to understand thestability switches of system (3), it is crucial to determine a threshold value of τ
at which (4) has a pair of conjugate pure imaginary roots. It is clear that λ = 0
is not a solution of (4). We then assume without loss of generality that λ = iυ,υ > 0, is a root of the transcendental equation and write the real and imaginaryparts as follow:
γ cos υτ = 0andυ − γ sin υτ = 0.

Adding up the squares of both equations yields υ2 = γ 2 from which we obtainυ = γ.
The characteristic equation is a function of the delay τ , and hence the roots ofthe characteristic equation are also functions of the delay. Diﬀerentiating thecharacteristic equation with respect to τ yields(1 − γτ e−λτ )dλ= γλe−λτ .
dτwhich is reduced toµ ¶−1
1 − γτ e−λτλdλ=
and from (4), e−λτ = − .
dτ
γλe−λτ
γ
Thus
Re

"µ

dλ
dτ

¶−1 #

=

1
> 0.
υ2

This inequality implies that all roots that cross the imaginary axis at iυ crossfrom left to right as τ increases. From the characteristic equation for λ = iυ,cos υτ = 0
andsin υτ =υ= 1.
γ

There is a unique υτ ∈ (0, 2π] such that υτ satisfies both equations, υτ = π/2.
We denote the threshold value of τ as
τ∗ =

π
1.57
'
.
2γ
γ

This result is summarized as follow:

4(5)

Theorem 1 System (3) has a threshold value τ ∗ of delay: the stationary point
is asymptotically stable when 0 < τ < τ ∗ and it is unstable when τ > τ ∗ where
τ∗ =

π
.
2γ

Theorem 1 deals with the case in which the monopoly firm uses a smallneighborhood of one point on the demand curve to determine its dynamic behavior. It implies that a small delay does not violate stability of the systembut a larger delay might destabilize it. We will next examine stability if themonopoly firm determines its rate of the output change by taking into accountrealized outputs at time periods, t − τ 1 > 0 and t − τ 2 > 0. The dynamic
equation having two time delays depends on the linear combination of the past
outputs,q̇(t) = α [a − c − 2b (wq(t − τ 1 ) + (1 − w)q(t − τ 2 ))](6)
where w (∈ (0, 1)) and 1 − w are the weights of the past outputs. We assume
that w ≥ 1 − w, that is, more weight is given to the more recent output1 . The
introduction of variable x(t) = q(t) − q M again reduces equation (6) toẋ(t) = −γ[wx(t − τ 1 ) + (1 − w)x(t − τ 2 )].

(7)

Looking for a solution x(t) = x0 eλt yields the characteristic equation
λ + γwe−λτ 1 + γ(1 − w)e−λτ 2 = 0.
For notational simplicity, letλ̄ =λ, γ 1 = γτ 1 and γ 2 = γτ 2γ

which then leads to
λ̄ + we−λ̄γ 1 + (1 − w)e−λ̄γ 2 = 0.
Dropping the bar from λ̄, we obtain the normalized characteristic equation
λ + we−λγ 1 + (1 − w)e−λγ 2 = 0.

(8)When γ 1 = γ 2 , equation (8) is essentially the same as equation (4). Thusγ 1 6= γ 2 is assumed henceforth.
Symmetric weights: w =1
2

1 Mathematically, the case of ω > 1 − ω is dual to the case of ω < 1 − ω. The results
obtained in one case reduces to the results in the other if ω and 1 − ω are replaced with 1 − ωand ω, respectively.

5Hale (1979) determines the geometry of the stability region for a linear differential equation with two delays when the weights of the delays are the same.
If λ = iυ in equation (8) with w = 1/2, then the real and imaginary parts satisfy
0=11
cos[υγ 1 ] + cos[υγ 2 ]
2
2

υ=

1
1
sin[υγ 1 ] + sin[υγ 2 ]22andor, equivalently,∙¸∙¸υ(γ 1 + γ 2 )υ(γ 1 − γ 2 )
cos
2
2
∙
¸
∙
¸
υ(γ 1 + γ 2 )υ(γ 1 − γ 2 )υ = sincos.
2
2
0 = cos

Notice that
cos

∙

(9)

¸
υ(γ 1 − γ 2 )
6 0,
=
2

otherwise the two equations of (9) do not hold simultaneously. Setting the firstequation of (9) equal to zero and solving it for υ produce the main value solution2υ=π
γ1 + γ2and from the second equation of (9),∙¸π(γ 1 − γ 2 )
π
= cos
.
γ1 + γ22(γ 1 + γ 2 )This is a hyperbolic curve passing through the point (π/2, π/2) and divides the(γ 1 , γ 2 ) space into two parts as illustrated in Figure 1. All roots of equation(8) have strictly negative real parts for (γ 1 , γ 2 ) in the gray region under the
curve, and instability of system (7) occurs for (γ 1 , γ 2 ) in the white region above
the curve. We call it a partition curve. It is also observed that the curve issymmetric with respect to the diagonal and asymptotic to the line γ i = 1,
which then implies that the stationary state is asymptotically stable for any
γ j > 0 if γ i ≤ 1 for i, j = 1, 2 and i 6= j. Comparing this last result with
Theorem 1 reveals that the two time delays increase the stability region when
the weights are the same.
2 System (9) has a general solutions

υ=(2k + 1)πγ1 + γ2and(2k + 1)ππ γ2 − γ1= cos (2k + 1)
γ1 + γ2
2 γ1 + γ2
The main value solution means the solution for k = 0.
(−1)k

6

k = 0, 1, 2, ...

Theorem 2 System (7) with w = 12 has a partition curve and its stationary
state is asymptotically stable for (γ 1 , γ 2 ) below the curve and unstable above it
where the partition curve is defined by the locus (γ 1 , γ 2 ) satisfying∙¸
π(γ 1 − γ 2 )
π
= cos
.
γ1 + γ22(γ 1 + γ 2 )Figure 1. Division of the (γ 1 , γ 2 ) space when ω =Asymmetric weights: ω >

1
2

1
2

Diﬀerential equations with two time delays are very diﬀerent from diﬀerential
equations with a single time delay when γ 1 6= γ 2 . Following the analysis of Haleand Huang (1993), we give a geometric description of the steady state for system(7) in the (γ 1 , γ 2 ) parameter space. We assume that λ = iυ with υ > 0 is a rootof equation (8) satisfyingiυ + ωe−iυγ 1 + (1 − ω)e−iυγ 2 = 0.
Let C denote the complex plane andΓ|1−ω| = {z ∈ C : |z| = |1 − ω|}.

Denote γ 1 by r, γ 2 by σ and υr = s for notational simplicity. For r > 0, definefunction fr : [0, 2π] → C bys
fr (s) = i + ωe−is .
rLetg(r, s) = r2 (|fr (s)|2 − |1 − ω|2 )= s2 − 2ωrs sin[s] + r2 (ω 2 − (1 − ω)2 )7

We can find the two combinations of r and s such thatg(r, s) = 0 and

dr
∂g/∂s
=−
= 0.
ds
∂g/∂r

Solving the second condition for r yieldsr=sω(sin[s] + s cos[s])which is substituted into the first condition to obtain¤£s2 ω 2 (s2 cos2 [s] − sin2 [s]) + (ω 2 − (1 − ω)2 ) = 0.

The graphical representation of this equation shows the existence of two solutions, s∗0 and s∗2 , and corresponding r0 and r2 ,
r0 =

s∗0
ω(sin[s∗0 ] + s∗0 cos[s∗0 ])

r2 =

s∗2
.
∗
ω(sin[s2 ] + s∗2 cos[s∗2 ])and

As can be seen in Figure 2,
fr ([0, 2π]) ∩ Γ|1−ω| 6= Ø
for r ∈ [r0 , r2 ]. The two dotted points in Figure 2 indicate that fri ([0, 2π]) ∩Γ|1−ω| = {fri (si )} for i = 0, 2. Since the fr ([0, 2π]) curve shifts downward as rincreases, there is a value r1 ∈ (r0 , r2 ) such that fr1 (s) passes through the point(−(1 − ω), 0), that is,£¤cos−1 − 1−ω
ω
£
¤
r1 =
ω sin[cos−1 − 1−ω]ωands2 (r1 ) = r1p
ω 2 − (1 − ω)2 .

Now for each r ∈ (r0 , r2 ), let pi = fr (s(r)) for r = 1, 2 and θi (r) ∈ [0, 2π]
denote the angles from the negative real axis to the rays starting at the origin
and passing through pi in the clockwise direction respectively as illustrated in8Figure 2.

Figure 2. Illustration of fr (s) and Γ1−w in C
Let α = Re[fr (si (r))] and β = Im[fr (si (r))]. Then we have explicit forms ofθ1 (r) and θ2 (r):
⎧
∙ ¸
⎪
−1 β
⎪
π − tanfor α > 0 and β ≶ 0,⎪⎪⎨α
θ1 (r) =
∙ ¸
⎪
⎪
⎪
−1 β
⎪
for α < 0 and β < 0,
⎩ 2π − tan
α

and

⎧
∙ ¸
⎪
−1 β
⎪
π − tanfor α > 0 and β > 0,
⎪
⎪
⎪
⎪
∙ ¸α
⎨
β
θ2 (r) =− tan−1
for α < 0 and β > 0,
⎪
α
⎪
∙ ¸
⎪
⎪
β
⎪
⎪
for α < 0 and β < 0.
⎩ 2π − tan−1αFunction θ1 (r) is continuous on [r0 , r2 ], and θ2 (r) is continuous on [r0 , r2 ]\{r1 }with a jump at r1 , sincelim θ2 (r) = 0 and

r→r1−

Now letσi (r) =lim θ2 (r) = 2π.

r→r1+rθi (r).
si (r)We then select the value w = 3/5 to have the red curve of σ = σ 1 (r) and theblue curve of σ = σ 2 (r) shown in Figure 3A in which the dotted point corresponds to (π/2, π/2) and the blue curve is discontinuous at r = r1 .3 Comparing
3 The shape of σ (r) depends on the value of w. Inequality σ (r) > σ (r) on (r , r ) can
1
2
1 2
i
be reversed for a larger value of w.

9Figure 2 with Figure 3A reveals the eﬀects on the stability region caused by
the asymmetric weights: the stability region is on the left side of the boundary
curves, σ = σ 1 (r) and σ = σ 2 (r). In Figure 3B, we periodically extend σ i (r) by
σ ni (r) = σ i (r) +2nπr
for r ∈ [r0 , r2 ] (i = 1, 2).
si (r)

Dynamic system (7) is locally stable for (r, σ) in the gray region and it is locallyasymptotically stable for any σ > 0 if r < r0 . Furthermore, the stability changes
as (r, σ) crosses the σ = σ i (r) locus (i.e., stability switch occurs).
Theorem 3 System (7) with ω > 12 is globally asymptotically stable for (r, σ)in the shaded region of Figure 3 and unstable otherwise.

B. Graph of σ ni (r) for i = 1.2.

A. Graphs of σ 1 (r) and σ 2 (r)Figure 3. Stability region of (7) with w =35Continuously distributed time delay is an alternative approach to deal withdelays. If the expected deviation of the output from its equilibrium value isdenoted by xε (t) at time t and is based on the entire history of the actual changesof the deviations from zero to t, then the gradient dynamics with continuously
distributed time lag can be written as the system of integro-diﬀerential equations⎧ẋ(t) = −γxε (t),
⎪
⎪
⎨
Z t
(10)
⎪
⎪
⎩ xε (t) =w(t − s, τ , m)x(s)ds,0

where the weighting function is defined by⎧ 1 t−s
− τ
⎪
⎪
⎨ τe
w(t − s, τ , m) =³ ´⎪⎪
⎩ 1 m m+1 (t − s)m e− m(t−s)
τ
m! τ
10

if m = 0,
if m ≥ 1.

Here m is a nonnegative integer and τ is a positive real parameter, which is associated with the length of the delay. To examine local dynamics of system (10)in the neighborhood of the stationary point, we substitute the second equationof (10) into the first to get
Z tẋ(t) + γw(t − s, τ , m)x(s)ds = 0.
0

Looking for the solution in the usual exponential form x(t) = x0 eλt and substituting it into the above equation, we obtainZ tw(t − s, τ , m)e−λ(t−s) ds = 0.
λ+γ0Introducing the new variable z = t − s simplifies the integral as
Z t
Z t−λ(t−s)w(t − s, τ , m)eds =w(z, τ , m)e−λz dz.
00

By letting t → ∞ and assuming that Re(λ) + m
τ > 0, we haveZ ∞1 − z −λzdz = (1 + λτ )−1 if m = 0
e τe
τ
0
and
Z ∞
0

That is,

1 ³ m ´m+1 m − mz −λz
z e τ e
dz =
m! τ
Z ∞

w(z, τ , m)e−λz dz =

0

with

`=

µ
¶−(m+1)λτ
1+
if m ≥ 1.
mµ¶−(m+1)λτ
1+`⎧⎨ 1 if ` = 0,⎩`

if ` ≥ 1.

Then the characteristic equations becomes³τ ´m+1
λ 1+ λ+ γ = 0.
`(11)

Expanding the characteristic equation presents the (m + 2)-th order equation
a0 λm+2 + a1 λm+1 + ... + am+1 λ + am+2 = 0
where the coeﬃcients ai are defined by³ τ ´m+1−k µ m + 1 ¶for 0 ≤ k ≤ m,
ak =
k
`
11

am+1 = 1 and am+2 = γ.
According to the Routh-Hurwitz stability condition, the necessary and suﬃcientconditions that all roots of the characteristic equation have negative real parts
are the following:(1) the coeﬃcients are positive, ak > 0 for k = 0, 1, 2, ..., m + 2,(2) the principle minors of the Routh-Hurwitz determinant are positive,¯¯¯¯¯ a1 a0 0 0 ¯¯¯¯¯ a1 a0 0 ¯¯¯¯¯ a a0 ¯¯¯¯ > 0, D3m = ¯ a3 a2 a1 ¯ > 0, D4m = ¯ a3 a2 a1 a0 ¯ > 0, ...
D2m = ¯¯ 1
¯
¯
¯
¯
¯
a3 a2
¯ a5 a4 a3 ¯¯ a5 a4 a3 a2 ¯¯ a7 a6 a5 a4 ¯Case 1. m = 0.

The characteristic equation (11) is quadratic, τ λ2 + λ + γ = 0. Since allcoeﬃcients are positive, the stationary state is locally asymptotically stable forall τ > 0, in which the delay is called harmless.
Case 2. m = 1.
The characteristic equation (11) is cubic and its coeﬃcients are a0 = τ 2 > 0,
a1 = 2τ > 0, a2 = 1 > 0, a3 = γ > 0 and D21 = τ (2 − τ γ). To obtain D21 > 0,
the delay τ should be less than the threshold valueτ ∗1 =

2
.
γ

(12)

Case 3. m = 2.
The characteristic equation (11) is 4th-order and its coeﬃcients are a0 =
τ 3 > 0, a1 = 6τ 2 > 0, a2 = 12τ > 0, a3 = 8 > 0, a4 = 8γ > 0, D22 = 64τ 3 > 0
and D32 = 32τ 3 (16 − 9γτ ). To obtain D33 > 0, the delay τ should be less than
the threshold value
16
1.78
τ ∗2 =
'
.
(13)
9γ
γ
Case 4. m = 3.
The characteristic equation (11) is 5th-order and its coeﬃcients are a0 =
τ 4 > 0, a1 = 12τ 3 > 0, a2 = 54τ 2 > 0, a3 = 108τ > 0, a4 = 81 > 0, a5 = 81γ >0, D23 = 540τ 5 > 0, D33 = 972(48 + γτ ) > 0 and D43 = −6561τ 6 (γ 2 τ 2 + 336γτ −576). To obtain D43 > 0, the delay τ should be less than the threshold value√24(5 2 − 7)
1.71
∗
τ3 =
'
.
(14)
γ
γ
The relations (5), (12), (13) and (14) define the partition curves of (γ, τ )that divide the (γ, τ ) space into stable and unstable parts. The three partition
curves for m = 1, 2, 3 and one partition curve for the fixed time lag are depicted12in Figure 4. It can be seen that all curves are hyperbolic and the partition curves
with the continuously distributed time lag are approaching the partition curve
obtained under the fixed time lag from above. In addition to the result in Case
I of m = 0, Figure 4 implies that the stable region becomes smaller as the valueof m increases and converges to the region defined by the fixed time delay whenm tends to infinity. The results obtained are natural if we notice the propertiesof the weighting function. For m ≥ 1, zero weight is assigned to the most recent
output, rising to maximum at t − s, and declining thereafter. As m increases,
the function becomes more peaked around t − s and tends to the Dirac delta
function. In consequence for suﬃciently large m, the weighting function may
be regarded as very close to the Dirac delta function and the dynamic behavior
under the continuously distributed time delay is very similar to the one under the
fixed time delay. We can explain this phenomenon mathematically by noticingthat the characteristic equation (11) of the continuously distributed case can be
written as1
1¶ = 0,λ+γµ¶m+1 µ
τλ
τλ
1+
1+
m
m
and as m → ∞, it converges to
λ + γe−λτ = 0
which is (3), the characteristic equation of the fixed time delay case. In short,
under continuously distributed time delay, although we comprehensively use thedelayed or past data of outputs, the stability domain is sensitive to the shapesof the weighting function. This result shows the sharp diﬀerence from the result
obtained in the discrete delay model of Ahmad et al (2004) in which delay
increases the stability region.

Figure 4. Stability regions

13

3

Delay Nonlinear Monopoly

In this section, we are concerned with cyclical behavior of delay monopoly, which
is not observed in the linear framework. To this end, we consider the case inwhich the adjustment speed is a positive function of the output. In particular,
we adopt the linear dependency:
Assumption 2. α(q) = αq with α > 0.
The gradient dynamics with discrete time lag τ > 0 under Assumption 2 isgiven byq̇(t)= α [a − c − 2bq(t − τ )] .
q(t)This implies that the monopoly firm adjusts its growth rate of the output proportional to a change in profit. It is rewritten asµ¶q(t − τ )q̇(t) = γ̄q(t) 1 −(15)
qM
where γ̄ = 2bαq M with q M = (a − c)/2b as before. This is the delayed logistic equation with one discrete delay, which is called the Hatchinson equation.
Introducing the new variabley(t) = −1 +

q(t)
,
qM

rescaling the time t = τ t̄ and letting y(t) = ȳ(t̄) reduce the dynamic equation(15) into the form·ȳ(t̄) = −τ γ̄ ȳ(t̄ − 1)(1 + ȳ(t̄)).

Dropping the bars from t̄ and ȳ and denoting β = τ γ̄ yieldẏ(t) = −βy(t − 1)(1 + y(t)).

(16)

We first turn attention to the stability of the zero solution y(t) = 0 of equation(16), which is equivalent to the stability of q(t) = q M in equation (15). Linearizing (16) at y(t) = 0 and letting yδ denote the deviation of y from its stationary
level giveẏδ = −βyδ (t − 1).

which is essentially the same as (3). Its corresponding characteristic equation is
λ + βe−λ = 0.
Following the same procedure taken above, we can show that the trivial solution
is asymptotically stable for β < π/2 (i.e., τ < π/2γ̄) and unstable for β > π/2(i.e., τ > π/2γ̄). Furthermore, if we assume that φ is a continuous function in
the interval [−1, 0], its value is at least −1 everywhere, but at zero it has to bestrictly greater than −1, then we have the following global stability result due
to Wright (1955)4 .
4 The following is Theorem 2.1 of Kuang (1993). Wright’s conjecture that the global stability holds for β < π/2 still remains open.

14

3, and y(t) be the solution of (16) with initial function
2γ̄
φ. Then limt→∞ y(t) = 0.

Theorem 4 Let τ ≤If the monopolist takes into account of the several delayed outputs, thedynamic equation isPnq̇(t) = αq(t) [a − c − 2b i=1 ω i q(t − τ i )]∙¸Pnq(t − τ i )= 2γ̄q(t) 1 − i=0 ω i
qM

(17)

where for the sake of simplicity the sum of the weight coeﬃcients, ω i , is assumed
to be equal to unitynXω i = 1.
i=0Denotingy(t) = −1 +

q(t)
qM

and letting ai = 2γ̄ω i , we see that the last expression of (17) can be rewrittenasnXẏ(t) = −(1 + y(t))
ai y(t − τ i )
(18)
i=0

which is exactly the same as the delay logistic equation studied by Kuang (1991)
with initial conditionx(θ) = φ(θ), θ ∈ [−τ , 0], φ ∈ C and φ(0) > −1,

where τ = max{τ i, i = 1, 2, ..., n}. Applying Kuang’s result (Corollay 4.1 ofKuang (1991)), we have the following:
Pn
Theorem 5 Consider (18) with the above initial conditions. If τ i=1 ai ≤ 1then limt→∞ y(t) = 0.
Theorems 4 and 5 are concerned with the global asymptotical stability ofthe trivial solutions in the delay equations (16) and (18). Further, we can numerically examine global behavior when the local stability condition is violated.
In Figure 5, we take the parameters values making the system locally unstableand perform simulations in the (y(t), ẏ(t)) space to show the birth of a limit
cycle. In Figure 5A, a single delay is τ = 1 and β = 1.8. In Figure 5B, we limit
the number of delays to two. Two delays are τ 1 = 2 and τ 2 = 2.5 and otherparameter values are w1 = 3/5, w2 = 2/5 and γ̄ = 1, which make a1 = w1 anda2 = w2 . In both simulations the trajectories starting at the same initial point15y(0) = 0.05 converge to limit cycles.

A. Single delay τ

B. Multiple delays τ 1 and τ 2

Figure 5. Birth of a limit cycle
The gradient dynamics with continuously distributed time lag is given by⎧q̇(t) = αq(t) [a − c − 2bq ε (t)]
⎪
⎪
⎪
⎪
⎨
Zt
(19)
⎪
ε
⎪
q
(t)
=
w(t
−
s,
τ
,
m)q(s)ds⎪⎪⎩0

where the weighting function is assumed to be the same as it was specified in theprevious section. To examine local dynamics of this system in a neighborhoodof the equilibrium point q(t) = q M , we need to consider the linearized version
where x(t) = q(t) − q M ,⎧ẋ(t) = −γ̄xε (t),
⎪
⎪
⎪
⎪
⎨
Zt
⎪
ε
⎪
⎪ x (t) = w(t − s, τ , m)x(s)ds.
⎪⎩0which has exactly the same form as the linear system if γ is replaced by γ̄(=γq M ). It then follows that the characteristic equations becomes
µ

λτ
λ 1+
`

¶m+1

+ γ̄ = 0.

When m = 0, the delay is harmless and the equilibrium point is locallyasymptotically stable for any τ > 0. When m = 1, it can be shown by applying

16

(12) that the parameters ai of the corresponding characteristic equation satisfies
the local stability condition a1 a2 − a0 a3 > 0 or
τ<

2
.
γ̄

We will show the possibility of the birth of limit cycles when loss of stability
occurs. According to the Hopf bifurcation theorem, we have to show that thefollowing two conditions are satisfied: (1) the characteristic equation of the
dynamic system has a pair of pure imaginary roots and no other roots with zeroreal parts; (2) the real parts of these roots vary with a bifurcation parameter.
The D21 = a1 a2 − a0 a3 = 0 line divides the parameter space into stable andunstable parts. Substituting a3 = a1 a2 /a0 into the characteristic equation andfactoring it yield(a1 + a0 λ)(a2 + a0 λ2 ) = 0.
We have therefore three characteristic roots: two are purely imaginary and one
is real and negative,
r
a2
1
a1
2
λ1,2 = ± − = ±i and λ3 = − = − < 0.
a0
τ
a0
τ
The first condition of the Hopf theorem is satisfied. Next we select the delay τ as
the bifurcation parameter and consider the roots of the characteristic equationas continuous function:
τ 2 λ(τ )3 + 2τ λ(τ )2 + λ(τ ) + γ̄ = 0
Diﬀerentiating it with respect to τ givesdλ2τ λ3 + 2λ2.
=− 2 2dτ
3τ λ + 4τ λ + 1
Substituting λ = i/τ , rationalizing the right hand side and noticing that theterms with λ and λ3 are imaginary and the constant and λ2 are real yield thefollowing form of the real part of the derivative of λ with respect to τ¸∙1dλ= 2 >0Reidτ |λ= τ5τwhich indicates that the second condition is also satisfied. Figure 6 illustrates
the birth of a limit cycle.5 We then summarize this result as follows:
5 The parameters are specified as a = 5, b = 1, c = 1 and τ = 0.8. The initian values of all
variables are the same, q M − 0.1. The dynamic system is⎧q̇(t) = −γ̄q e (t),
⎪
⎪
⎪
⎨ e
1
q̇ (t) = (x(t) − q e (t)),
τ
⎪
⎪
1
⎪
⎩ ẋ(t) = (q(t) − q e (t)),τ

17Theorem 6 The monopolistic equilibrium point of the continuously distributed
delay system (19) with m = 1 is destabilized through a Hopf bifurcation when
the delay τ crosses the critical valueτ̄ ∗1 =

2.
γ̄Figure 6. Birth of a limit cycle when m = 1
In the same way, we can show the existence of a limit cycle when m = 2.
In particular, (13) implies that τ̄ ∗2 = 16/(9γ̄) is the threshold value of the delay
and from (Case 3), the curve of a1 a2 a3 − (a0 a23 + a21 a4 ) = 0 is the partition
curve between the stable and unstable regions. Solving the partition curve fora4 , substituting it into the characteristic equation and factoring the resultant
equation yield(a3 + a1 λ2 )(a1 a2 − a0 a3 + a21 λ + a0 a1 λ2 ) = 0.
Two of the characteristic roots are purely imaginary,ra32
λ1,2 = ± − = ±i √a1
τ 3
and the other two roots are the solutions of the quadratic equation (a1 a2 −a0 a3 ) + a21 λ + a0 a1 λ2 = 0 and have negative real parts, since all coeﬃcients are
where

t

q e (t) =
0

1
τ

and

2

t 1

x(t) =
0t−s(t − s)e− τ q(s)dsτt−se− τ q(s)ds.

18

positive. Diﬀerentiating the characteristic equation with respect to τ givesdλ3τ 2 λ4 + 12τ λ3 + 12λ2.
=− 3 3dτ
4τ λ + 18τ 2 λ2 + 24τ λ + 8
2and taking the real part, we have
Substituting λ = i τ √
3

"

dλ
Re
dτ |λ=i τ √2 3

#

=

60
> 0.
439τ 2

Theorem 7 The monopolistic equilibrium point of the continuously distributed
delay system (19) with m = 2 is destabilized through a Hopf bifurcation when
the delay τ crosses the critical valueτ̄ ∗2 =

4

16
.
9γ̄

Concluding Remarks

This paper presents a delay monopoly model with linear price and cost functionand examines its stability when dynamics is driven by the gradient process. The
paper is divided into two parts and systems dynamics are compared with fixedand continuously distributed time delays in both parts. In the first part, we
assume constant speed of adjustment and detect the threshold value of delay
at which stability switch occurs. The monopoly with the single fixed delay has
the minimum stable region while the continuously distributed delay becomes
harmless when its weights are declining with the most weight given to the mostcurrent output. The stability region under the continuously distributed timedelay converges to the stability region under the single fixed time delay as theweight function is approaching the Dirac delta function. It is confirmed thatthe stability region becomes complex when there are two fixed time delays. In
the second part, we assume output-dependent speed of adjustment. This replacement of the adjustment speed makes the dynamic system nonlinear(i.e., alogistic equation). Local and global stability conditions are derived. Furthermore, the possibility of the birth of limit cycles is demonstrated when local
stability is violated.

19

References[1] Ahmad, E., H. Agiza and S. Hassen, "On Modifications of Puu’s Dynamical
Duopoly," Chaos Solitions and Fractals, 11(2000), 1025-1028.
[2] Bélair, J., and M. Mackey, "Consumer Memory and Price Fluctuations
in Commodity Markets: An Integrodiﬀerential Model," Journal of Dynamics and Diﬀerential Equations, 1 (1989), 299-325.
[3] Bischi, G. I. and F. Lamantia, "Chaos Synchronization and Intermittency
in a Duopoly Game with Sprillover Eﬀects," in Oligopoly Dynamics, T.
Puu and I. Sushko (eds), 195-217, 2002, Berlin/Heidelberg/New York,
Springer-Verlag.
[4] Bischi, G. I. and A. Naimuzada, "Global Analysis of a Duopoly Game with
Bounded Rationality," Advances in Dynamic Games and Applications,
5 (1999), 361-385.
[5] Chiarella, C., and A. Khomin, "An Analysis of the Complex Dynamic Behavior of Nonlinear Oligopoly Models with Time Lags," Chaos, Solitons
and Fractals, 7 (1996), 2049-2066.
[6] Chiarella, C., and F. Szidarovszky, "The Birth of Limit Cycles in Nonlinear Oligopolies with Continuously Distributed Information Lags," inModeling Uncertainty, Dror, M., P. L’Ecuyer and F. Szidarovszky (eds),
249-268, 2001, Boston, Kluwer Academic Publisher.
[7] Hale, J., "Nonlinear Oscillations in Equations with Delays," in Nonlinear
Oscillations in Biology, K. C. Hoppenstadt (ed), Lectures in Applied
Mathematics, 17 (1979), 157-185.
[8] Hale, J., and W. Huang, "Global Geometry of the Stable Regions for Two
Delay Diﬀerential Equations," Journal of Mathematical Analysis andApplications, 178(1993), 344-362.
[9] Invernizzi, S., and A. Medio, "On Lags and Chaos in Economic Dynamic
Models," Journal of Mathematical Economics, 20 (1991), 521-550.
[10] Kuang, Y., Delay Diﬀerential Equations with Applications in Population
Dynamics, Boston, Academic Press, 1993.
[11] Puu, T. Attractor, Bifurcations and Chaos, 2003, Berlin/Heidelberg/New
York, Springer-Verlag[12] An der Heiden, U., and M. Mackey, "The Dynamics of Production and Destruction: Analytic Insight into Complex Behavior," Journal of Mathematical Biology, 16 (1982), 75-101.

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