Dynamic Monopoly with Multiple Continuously
Distributed Time Delays∗
Akio Matsumoto†

Ferenc Szidarovszky‡

Abstract
Two time delays are assumed in a boundedly rational monopoly. The
characteristic equation is derived for the general case, and a complete
stability analysis is conducted both analytically and numerically in two
special cases. In the first case in which the continuously distributed timedelays have different weights, it is shown that a monopoly equilibrium isdestabilized to generate a limit cycle via Hopf bifurcation. In the second case in which one delay is continuously distributed and the other isfixed, it is demonstrated that the stability of the monopoly equilibrium
can change finite number of times and eventually becomes periodicallyor aperiodically unstable. It is of interest to notice that the two delay
dynamics can be qualitatively different from the one delay dynamics.

Key words: Boundedly rational, Continuously distributed time delay,
Fixed time delay, Hopf bifurcation, Complex dynamics, Gradient dynamics
JEL Classification: C62, C63, D21, D42∗ The authors highly appreciate financial supports from the Japan Society for the Promotionof Science (Grant-in-Aid for Scientific Research (C) 24530202) and Chuo University (Grant
for Special Research and Joint Research Grant 0981).
† Department of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo, 1920393, Japan. akiom@tamacc.chuo-u.ac.jp‡ Department of Systems and Industrial Engineering, University of Arizona, Tucson, 857210020, USA. szidar@sie.arizona.edu

11Introduction

This study analyzes a dynamic model of a boundedly rational monopoly incontinuous-time scale and investigates the destabilizing effects caused by timedelay arisen in a process of collecting information on the demand function. It isa continuation of Matsumoto and Szidarovszky (2012b) in which the dynamic
monopoly model is constructed under a single continuously distributed time delay (continuous delay henceforth). The monopoly stationary point is shown tobifurcate to a limit cycle through Hopf bifurcation when it loses stability. It isalso a complement of Matsumoto and Szidarovszky (2012a) in which the continuous delay is replaced with a fixed time delay (fixed delay henceforth). It isdemonstrated that Hopf cycles emerge under one fixed delay and so does thecomplex dynamics involving chaos under two fixed delays. The point is that periodic behavior emerges if the quantity adjustment process has one time delay,
regardless of the delay being continuous or fixed, while aperiodic behavior canbe achieved if it contains two fixed delays. The natural inference from theseresults is that erratic behavior can be expected if multiple continuous delays are
involved. The main purpose of this study is to confirm this fact.
The principal impetus is provided by the dynamic analysis of the boundedly
rational monopoly with discrete-time scale conducted by Puu (1995) and, more
recently, Naimzada and Ricchiuti (2008). A gradient rule is assumed, in bothstudies, to determine production in such a way that production is increased
if a change in profit is positive, decreased if negative and constant if zero. A
cubic demand function with an inflection point is assumed in the former andthis particular nonlinearity is shown to be a main source for chaotic attractor.
On the other hand, a cubic demand without an inflection point is assumed in
the latter and stability is violated to chaos through the familiar period-doublingcascade. In this study, the same gradient dynamics is considered under different
conditions, namely, the demand function is linear, a continuous-time scale isadopted and the growth rate of output is proportional to the marginal change
in expected profit. Special attention is given to the destabilizing effect causedby two continuous delays.
The paper is organized as follows. Section 2 presents a basic monopoly modelwith continuous delays. Section 3 examines the case in which two continuous
delays have different weighting functions. Section 4 investigates the special casein which continuous and fixed delays coexist. Finally concluding remarks are
given in Section 5.

2

Delay Monopoly

2.1

Basic Model

Consider an output decision problem of a boundedly rational monopolistic firm
which produces output q with marginal cost c. The price function is assumed2

to be linear
f (q) = a − bq, a, b > 0.

It is further assumed that the firm does not want to react to sudden market
changes, so instead of the most current price information, an average of past
prices is used in the adjustment process. Because of the linearity of the pricefunction it is equivalent to the use of an average of past output data q e in
the adjustment scheme. Then the corresponding marginal profit is given asa − c − 2bq e which generates the approximating gradient dynamicsq̇(t)= α (a − c − 2bq e (t))
q(t)

(1)

with α > 0 being an adjustment coefficinet, furthermore t denotes a point ofcontinuous time and the dot over a variable means a time derivative. (1) implies
that the growth rate of output is adjusted in proportion to the average marginal
profit. In constructing best response dynamics, global information is required
about the profit function, however, in applying gradient dynamics, only local
information is needed. The dynamic equation is written asq̇(t) = αq(t) [a − c − 2bq e (t)](2)

Since q(t) = q e (t) holds at a stationary point, equation (2) has two stationary
points; the zero trivial point q(t) = 0 and a nontrivial point
qM =

a−c
2b

where a > c is assumed to ensure that the nontrivial point is positive. We callq M a monopoly equilibrium. Dynamic behavior of (2) depends on the formationof averaging past data. With continuous-time scale, time delays can be modeledwith a continuous or fixed delay. As is mentioned in the Introduction, dynamic
analysis has been done under the single continuous delay, one and two fixed
delays. In this study we adopt multiple continuous delays and draw attention
to the destabilizing effects of continuous delays having different weights. In
addition we will investigate the limiting case when one delay is continuous andthe other is fixed.

2.2

Continuous Delays

In our economic situation with continuous delays, the monopolistic firm gathers information about the actual demands in the past and forms an average by
weighting the different transactions according to their likelihood. One particular rule is that weights are exponentially declining with the most weight given to
the most current transaction. If the largest weight is given to some past transaction, an appropriate formation rule is that small weight is given to the mostcurrent data, rising to maximum at the particular date and declining thereafter.
According to the latter rule, weights take a bell-shaped form. We draw attention to the special case in which the firm uses the combination of these different
3rules to estimate the incoming demand. To keep the rule simple and tractable,
we assume that the firm forms its average with two steps: at the first step thetwo weighted averages of the past data, q1ε (t) and q2ε (t), are calculated basedon the entire history of the same actual output values with different weights;
at the second step, the average demand is determined as the weighted averageof q1ε (t) and q2ε (t). Since this rule uses the two continuous delays having different weights, we call it the formation rule with heterogeneous weights or simply
heterogeneous weights.1 The gradient dynamics with heterogeneous weights iswritten as the system of the Volterra type integro-differential equations2q̇(t) = αq(t) [a − c − 2b (ωq1ε (t) + (1 − ω)q2ε (t))] ,








t



ε

 q1 (t) = W (t − s, S, m)q(s)ds,
(3)
0






t



ε

q (t) = W (t − s, T, n)q(s)ds,


 2
0

where 0 < ω < 1, m and n are nonnegative integers, T and S are positive realparameters. The weighting function is defined by 1 t−s− τif ℓ = 0,

 τe

(4)
W (t − s, τ , ℓ) = ℓ+1ℓ(t−s)1 ℓ(t − s)ℓ e− τ
if ℓ ≥ 1ℓ! τfor τ = S, T and ℓ = m, n. Parameter τ is associated with the average lengthof the continuous delay and parameter ℓ determines the shape of the weighting
function. For ℓ = 0, weights are exponentially declining. For ℓ ≥ 1, the shapeof the weighting function takes a bell-shaped form which becomes taller andthinner as ℓ increases.
We linearize the dynamic system (3) in a neighborhood of the monopoly
equilibrium, to examine local stability. If the actual and average deviations of
the output from their equilibrium value at time t are denoted by qδ (t) = q(t)−q M

1 This hybrid rule can arise in a cartel-monopoly situation in which cartel members havedifferent likelihood on the past transactions but agree on a unique pricing.
2 If the average demand is equal to the fixed delay demand, then the dynamic system
can be transformed to a system of delay differential equations, which is rigorously studied in
Matsumoto and Szidarovszky (2012b).

4

εand qδ,i(t) = qiε (t) − q M for i = 1, 2, then linearized version of (3) has the form


ε
ε

(t)
=
−γ
ωqδ,1
(t) + (1 − ω)qδ,2
(t) ,
q̇

δ






 t


ε
(t) =
W (t − s, S, m)qδ (s)ds,
qδ,1
(5)
0





 t


 ε

 qδ,2 (t) =W (t − s, T, n)qδ (s)ds0with γ = 2αbq M . Substituting the second and third equations of (5) into the first
and then substituting the exponential form of the solution, qδ (t) = eλt u, into
the resultant equation present the following form of the characteristic equation t t−λ(t−s)λ = −γ ωW (t − s, S, m)e
ds + (1 − ω)W (t − s, T, n)e−λ(t−s) ds .
00Introducing the new integration variable t − s = z yields
 t t
W (z, S, m)e−λz dz + (1 − ω)W (z, T, n)e−λz dz .
λ = −γ ω00

By letting t → ∞, we have ∞0

(1 + λτ )−1 if ℓ = 0,
−(ℓ+1)W (z, τ , ℓ)e−λz dz =

λτ

 1+
if ℓ ≥ 1,ℓwith τ = S, T and ℓ = m, n. The usual form of the characteristic equation istherefore−(m+1)
−(n+1)λT
λS(6)+ (1 − ω) 1 +λ = −γ ω 1 +
m̄
n̄
where
m̄ =


 1 if m = 0,and n̄ =m if m ≥ 1,
 1 if n = 0,n if n ≥ 1.

Notice that equation (6) can be rewritten as a polynomial equation m+1 

n+1
m+1

λT n+1=0
λ 1 + λS1++γ
ω 1 + λT+ (1 − ω) 1 + λS
m̄
n̄
n̄
m̄
(7)
showing that the spectrum is finite with m + n + 3 eigenvalues. This is thegeneral form of the characteristic equation with two different continuous delays.
It is, however, not easy to derive general stability conditions, so we focus onsome special cases with small values of m and n.
5

Before proceeding, we deal with two cases: the no-delay case as a benchmark
and the harmless-delay case. First substituting S = T = 0 into equation (7)yields the following form of the characteristic equation,
λ + γ [ω + (1 − ω)] = 0or
λ = −γ = −α(a − c) < 0.
The last inequality implies the local asymptotic stability of the monopoly equilibrium since the eigenvalue is real and negative. We next examine the casewith m = n = 0 in which both delays have exponentially declining weights. The
characteristic equation (7) can be written as
λ(1 + λS)(1 + λT ) + γω(1 + λT ) + γ(1 − ω)(1 + λS) = 0or

b0 λ3 + b1 λ2 + b2 λ + b3 = 0

(8)

with
b0 = ST, b1 = S + T, b2 = 1 + γωT + γ(1 − ω)S and b3 = γ.
All coefficients are positive. According to the Routh-Hurwitz stability criterion,
equation (8) has roots only with negative real parts if and only if
b1 b2 − b0 b3 = (S + T )(1 + γωT + γ(1 − ω)S) − ST γ > 0
where the middle expression is re-written as
S 2 γ(1 − ω) + T 2 γω + S + T
which is always positive. Hence the monopoly equilibrium is locally asymptotically stable regardless of the values of S and T . In other words, the delays
become harmless when the weights are exponentially declining. Summarizingthese results, we have the following:
Proposition 1 The monopoly equilibrium is locally asymptotically stable if nodelays exist or if the two continuous delays have exponentially declining weighting functions.

3

Heterogeneous Weights: m = 0 and n ≥ 1

We now examine the case of m = 0 and n ≥ 1 in which delay T has a bell-shaped
weight while delay S has a declining weight. Substituting m = 0 reduces thecharacteristic equation (7) to
n+1
n+1


T
T
+γ ω 1+ λ+ (1 − ω) (1 + λS) = 0. (9)λ (1 + λS) 1 + λnn6With expanding the factored terms, this can be reduced to a polynomial equation of degree n + 3,
b0 λn+3 + b1 λn+2 + · · · + bn+2 λ + bn+3 = 0(10)where the coefficients are defined by
b0 = a0 S,
b1 = a1 S + a0 ,bk = ak S + ak−1 + γωak−1 for 2 ≤ k ≤ n + 1,
bn+2 = an+1 + γωan + r(1 − ω)S,
bn+3 = γwithak = n+1−k S
m+1
for 0 ≤ k ≤ n + 1.
kn

In the case of polynomial equations, the Routh-Hurwitz theorem3 provides
the necessary and sufficient conditions for all the roots to have negative realparts. Applying the theorem, we first construct the (n + 3) × (n + 3) RouthHurwitz determinant:0
b1 b0 0 0 · · · b3 b2 b1 b0 · · ·0  b5 b4 b3 b2 · · ·0 .
Dn+3 = det 0  b7 b6 b5 b4 · · ·
 ·
·
·
· ···
0 
0 0 0 00 bn+3We then check whether the following two conditions are satisfied:(S1) all coefficients are positive, bk > 0 for k = 0, 1, 2, ..., n + 3,
(S2) the principal minors of the Routh-Hurwitz determinant are all positive,
n+2
2
3
Dn+3
> 0, Dn+3> 0, ... Dn+3>0
n+2kwhere Dn+3
is the k-th order leading principal minor of Dn+3 and Dn+3
>
n+3
n+2
0 always leads to Dn+3 = γDn+3 > 0 since γ > 0.

We will investigate local stability in Section 3.1 and then consider global
stability in Section 3.2 that is further divided into three parts: the occurrenceof Hopf bifurcation is shown in the first part, this analytical result is numericallyconfirmed in the second part and the destabilizing effect caused by increasing
the value of n is examined in the third part.
3 See, for example, Gandolfo (2009) for this theorem. It is mentioned there that the LiénardChipart conditions have an advantage over the Routh-Hurwitze conditions because the former
involves about half as many determinantal inequlities as the latter.

7

3.1

Local Stability

The characteristic equation (10) with n = 1 becomes quartic,b0 λ4 + b1 λ3 + b2 λ2 + b3 λ + b4 = 0
with

(11)

b0 = ST 2 ,b1 = 2ST + T 2 ,b2 = S + 2T + γωT 2 ,b3 = 1 + γ(1 − ω)S + 2γωT,
b4 = γ.

Since the coefficients are all positive (i.e., condition (S1) is satisfied), the stability condition that all the roots of the characteristic equation have negative
real parts is validated if
b1 b0 0b1 b0
> 0 and D43 = det  b3 b2 b1  > 0.
D42 = det
b3 b2
0 b4 b3

Clearly D43 > 0 implies D42 > 0. Therefore the locus of D43 = 0 determines thestability region in case of n = 1. In particular, the two loci of D42 = 0 and
D43 = 0 are depicted as C-shaped dotted and solid curves in Figure 1(A) where
we take γ = 1 and ω = 1/2. The inequalities D43 > 0 and D42 > 0 hold to theleft of the corresponding curves and they are reversed to the right. The former
curve is located in the region where D43 < 0, implying that D42 > 0 always in theregion where D43 > 0. The D43 = 0 locus partitions the nonnegative (S, T ) plane
into a stable (yellow) region and unstable (white) region and thus is called apartition curve.
The shape of the D43 = 0 locus indicates that equation D43 = 0 has equalroots at the turning point of the solid curve. Checking the existence of equalroots, we expand D43 to obtain a quartic polynomialf(T ) = aT 4 + bT 3 + cT 2 + dT + d
where the coefficients are given by
a = 2ω 2 γ 2 ,b = 2(1 − 2Sγ(1 − 2ω)),c = 2Sγ(1 − Sγω),d = 4S − (1 − ω)S 2 − (1 − ω)2 γ 2 S 3 ,e = 2(1 + Sγ(1 − ω))S 2 .
8

It is known that the resultant with respect to f (T ) and f ′ (T ) is equal to −Ω
where Ω is the discriminant of f (T ),


a b
c
d e 0 0 0
a bcd e 0 


 0
0
a b
c d e ′Res(f, f ) = det  4a 3b 2c d 0 0 0  = −Ω.
 0
4a 3b 2c d 0 0 


 0
0
4a 3b 2c d 0 000
4a 3b 2c d

It is not easy to derive a general form of the equal roots. However, it can becalculated if the parameters are specified. Under the parameter setting (a =2, b = c = 1, α = 1, ω = 1/2), quartic polynomial f(T ) has equal roots at point
C with S0 ≃ 8.97 and T0 ≃ 4.72 in Figure 1(A).4 Then it can be mentionedthat the monopoly equilibrium is locally asymptotically stable regardless of thevalues of T if S < S0 . On the other hand, f(T ) has two distinct solutions for
S > S0 . In Figure 1(A), the vertical line at S = S1 crosses the partition curveat two points, A and B, whose coordinates are TA and TB , respectively. The
monopoly equilibrium loses stability at point A and regains stability at point B
when T increases along the vertical line S = S1 . Figure 1(B) is the bifurcation
diagram with respect to T and depicts that a limit cycle is generated for T in
the interval (TA , TB ) in which the monopoly equilibrium is locally unstable. We
will work out the details of this bifurcation diagram and the birth of such cyclic
behavior in the next subsection.

(A) Stability region(B) Bifurcation diagram

Figure 1. Heterogeneous weights with m = 0 and n = 1
4 This calculation and the following numerical simulations are done with Mathematica,
version 7.

9

3.2

Global Stability

3.2.1

Hopf Bifurcation

The next question which is naturally arisen is on global dynamics when theequilibrium loses stability. To investigate such dynamics, we will analyticallyshow that a Hopf bifurcation occurs on the partition curve and then numericallyconfirm the emergence of a limit cycle when the monopoly equilibrium is locallyunstable. In order to apply the Hopf bifurcation theorem,5 we have to check
whether the following two conditions are satisfied:(H1) the characteristic equation has a pair of purely imaginary roots and has
no other roots with zero real parts;(H2) the real part of these roots vary with a bifurcation parameter.
We start with the first condition. The quartic characteristic equation canbe factored as follows when D43 = b1 b2 b3 − b20 b3 − b21 b4 = 0:
(b3 + b1 λ2 )(b1 b2 − b0 b3 + b21 λ + b0 b1 λ2 ) = 0.
It is then clear that the equation has a pair of purely imaginary roots and tworoots with no-zero real parts. In particular, the two purely imaginary roots areb3
λ1,2 = ± − = ±iβ
b1
with
β=



1 + γ(1 − ω)S + 2γωT
T (2S + T )and the other two roots are
λ3,4 =−b21 ±

b41 − 4b0 b1 (b1 b2 − b0 b3 )2b1 b2 − b0 b3

withRe [λ3,4 ] = 0.
Thus the first condition (H1) is shown to be satisfied.
We turn to verify (H2). Selecting T as a bifurcation parameter, we might
treat the characteristic root as a continuous function of T. Then differentiating
the characteristic equation (11) with respect to T and arranging terms, we
obtainST λ4 + 2T (1 + S)λ3 + 2(1 + γωT )λ2 + 2γ(1 − ω)λ
dλ
=−
.
dT
4b0 λ3 + 3b1 λ2 + 2b1 λ + b3
5 See, for example, Gandolfo (2009) for this theorem.

10At λ = iβ, the real part of the derivative is obtained asRedλ
2β 2 {(ST β 2 −(1+γωT ))(b3 −3b1 β 2 )+2(γω−(S+T )β 2 )(b2 −2b0 β 2 )}=−.
(b3 −3b1 β 2 )2 +4β 2 (b2 −2b0 β 2 )2
dT

(12)

Although the right hand side of (12) has the complicated form, we can numerically check its sign. Given the same setting of the parameters as in Figure 1,
the coordinates of points A and B for S1 = 20 are calculated as TA ≃ 3.35 andTB ≃ 17.63, respectively. Substituting these values into the right hand side ofequation (12) shows that it is approximately 0.0156 at point A and -0.0015 atpoint B. Hence we have


dλ 
dλ 
Re
> 0 at point A and Re< 0 at point B.
dT λ=iβdT λ=iβThese inequalities imply the fulfillment of the second condition (H2). Hencethe equilibrium point loses stability and bifurcates to a limit cycle at point A
while the limit cycle merges to the monopoly equilibrium at point B at which
stability is regained. It is further possible to demonstrate that the real partof the derivative is positive at any points on the downward-sloping part of theD43 = 0 curve and negative on the upward-sloping part in Figure 1(A). The
switching of the stability occurs twice when T increases and S is fixed to belarger than S0 .
3.2.2

Numerical Simulations

In this subsection, we numerically examine the stability switch and global behavior of the unstable equilibrium. The dynamic system under the investigationis constructed by substituting m = 0 and n = 1 into (3),q̇(t) = αq(t) [a − c − 2b(ωq1ε (t) + (1 − ω)q2ε (t))] ,q1ε (t) = t

q2ε (t) = t  2
1t−s(t − s)e− T q(s)ds.
T0

1 − t−se S q(s)ds,
S
0

Differentiating the second and third equations with respect to t and introducing
a new variable t
1 − t−sq0 (t) =e T q(s)dsT011

reduce the dynamic system with two continuously distributed time delays to a
4D system of the ordinary differential equations,q̇(t) = αq(t) [a − c − 2b(ωq1ε (t) + (1 − ω)q2ε (t))] ,








1


q̇1ε (t) = (q(t) − q1ε (t)) ,


S

1



q̇2ε (t) = (q0 (t) − q2ε (t)) ,


T







 q̇0 (t) = 1 (q(t) − q0 (t)) .
T

We use the same parameters setting as before, take initial values, q(0) = q1ε (0) =q2ε (0) = q0 (0) = q M − 0.2 and then perform simulation of the 4D system fordifferent values of T . Notice that γ = 1 under this parameter specification.
Setting S1 = 20 and increasing the value of T with the increment of 0.01 from0 to 25 along the vertical line S = S1 , we obtain the bifurcation diagram as
illustrated in Figure 1(B). For each value of T, the dynamic system is simulated
for 10, 000 iterations. The first 9, 950 are discarded and the local minimum andmaximum of the remaining 50 data are plotted vertically above the point T.
This leaves transient changes out of the picture and gives an estimate of global
behavior after initial disturbances. The output trajectory q(t) converges to themonopoly equilibrium for relatively smaller values, T < TA . It loses stability at
T = TA (i.e., point A) and bifurcates to a limit cycle for values up to TB . The
amplitude of the cycle first increases and then decreases as T increases fromTA to TB . The trajectory regains stability at T = TB (i.e., point B) and thenstays at the equilibrium for T > TB . Essentially the same phenomenon can beobserved for any S > S0 .
Proposition 2 Assume heterogeneous weights with m = 0 and n = 1. Increasing the value of T from zero and fixing S > S0 lead to the followings: (1) two
critical values TA (S) and TB (S) are obtained by solving D43 = 0 for T, given
S; (2) a limit cycle is born via Hopf bifurcation when the equilibrium loses stability at TA (S); (3) the cycle expands, shrinks and then finally merges to theequilibrium point when it regains stability at TB (S).
3.2.3

Stability Sensitivity to the Value of n

We have demonstrated that stability of the monopoly equilibrium is sensitive
to the length of the delay T , given S > S0 . We now draw attention to thevalue of n to see how the form of the weighting function affects dynamics. As
it is clear from (4), the shape of the weighting function depends on the valueof the shape parameter n. A larger n means graphically that higher weights
are concentrated to a smaller neighborhood of the maximum point t − T . The
degree of the characteristic equation becomes higher and so does the order ofthe leading principal minors of the corresponding Routh-Hurwitz determinant.
12

It then follows that the stability conditions become increasingly untractable.
However we can numerically check the stability conditions. Taking the samen+2
parameter setting, we illustrate the C-shaped partition curves of Dn+3
= 0 for6n = 1, 2, 3, 4, 5, 6, 7 in Figure 2(A). The right most curve corresponds to thecase of n = 1, and the left most curve to n = 7, implying that the partition
curve shifts leftward as n increases. It is also confirmed that the monopoly
equilibrium is locally stable to the left of the corresponding partition curve andunstable to the right. The boundaries of the instability regions are the partition
curves painted in black with the interior painted in gray colors. The gray color
becomes darker as n increases. Thus the most light-gray region corresponds tothe instability region with n = 1. The darker gray region surrounded by theright most curve and the next right most curve is added to it to construct the
instability region with n = 2. The instability regions with n ≥ 3 are obtained in
the same way, implying that the instability region becomes larger as n increases.
We present the bifurcation diagrams for n = 1, 2, 3, 4 on top of one another inFigure 2(B) where S = 10. It can be seen there that an inverse U -shaped part
becomes taller and wider with increasing n. This means that the amplitude ofthe limit cycle becomes larger and the instability interval defined as (TAn , TBn )n+2= 0, given S =becomes longer where Tkn for k = A, B is a solution of Dn+3
7
10. Replacing T for S, n for m and ω for 1 − ω, we can derive the partition
curves in the case of n = 0 and m ≥ 1. One case with n ≥ 1 and m = 0 can betransformed into the other case with m ≥ 1 and n = 0 by changing variables.
This means that the qualitative properties to be obtained do not depend on
the choice of fixed and changing variables. So we confine our attention, in thesequel, to the case where m is fixed and n is increased. We have seen thefollowing result:
Proposition 3 Assume heterogeneous weights with m = 0 and n ≥ 1. Larger
values of n have stronger destabilizing effects such that the amplitude of the limit
cycle becomes larger and the unstable interval (TAn , TBn ) becomes longer.
6 For n ≤ 7, it is numerically confirmed, as we did in Section 3-1, that D n+2 > 0 is sufficientn+3
k> 0 for k = n + 1, ..., 2.
to have Dn+3
7 Since T n has almost the same values for different value of n, we label only T 1 on theAAhorizontal line in Figure 2(B).

13

(A) The partition curves(B) The bifurcation diagrams

Figure 2. Heterogeneous weights with n = 1, 2, ...7 and m = 0.

4

Heterogeneous Delays: m = 0 and n → ∞The expanding instability effect caused by the increasing value of n is numerically confirmed. We now address ourself to an extreme case in which n goes toinfinity and m still remains at zero. Since we have
n 


λT
λT
= eλT ,
1+
lim 1 +n→∞nn
the weighting function converges to the Dirac delta function and so does thecontinuous delay to a fixed delay. The dynamic system (3) turns to be a differential equation with qualitatively different delays, one is a continuous delay
and the other is a fixed delay:
q̇(t) = αq(t) [a − c − 2b(ωq1ε (t) + (1 − ω)q2ε (t))] ,






 t

1 − t−sq1ε (t) =e S q(s)ds,

S

0




 ε
q2 (t) = q(t − T ).

We call this scheme the formation rule with heterogeneous delays or simply
heterogeneous delays. Substituting the third equation into the first and differentiating the second equation with respect to t, we obtain the following 2D
system of a delay differential equation and an ordinary differential equation:ε q̇(t) = αq(t) [a − c − 2b(ωq1 (t) + (1 − ω)q(t − T ))](13) q̇ ε (t) = 1 [q(t) − q ε (t)] .
1
1
S
14

Correspondingly the characteristic equation (7) is reduced toλ(1 + λS) + γω + γ(1 − ω)(1 + λS)e−λT = 0(14)

λ2 + a1 λ + a2 + (b1 + b2 λ)e−λT = 0

(15)

or

where the coefficients are
a1 =

1
γω
γ(1 − ω), a2 =, b1 =
and b2 = γ(1 − ω).
SSS

It has been shown that the monopoly equilibrium is locally stable for T =0. We are concerned with whether the stability of the equilibrium switches toinstability even under heterogeneous delays when T increases. If such stability
switch occurs at T = T̄ , then the characteristic equation (15) must have a pairof purely imaginary conjugate roots. The central problem of the stability switch
is to determine this critical value of T̄ . To this end, we suppose without lossof generality that λ = iυ, υ > 0 is a root of (15) for T = T̄ and rewrite thecharacteristic equation as a set of two equations making the real part and theimaginary part equal to zero,a2 − υ2 + b1 cos υT + b2 υ sin υT = 0,(16)

a1 υ + b2 υ cos υT − b1 sin υT = 0.
Moving a2 −υ 2 and a1 υ to the right hand sides of the equations in (16), squaring
them and adding them yieldb21 + b22 υ 2 = (υ2 − a2 )2 + a21 υ 2 .
Hence
υ4 − Aυ 2 + B = 0

(17)

where
A = b22 + 2a2 − a21and
B = a22 − b21 =γ2(2ω − 1).
S2

Solving (17) for υ 2 gives two rootsA±υ 21,2 =√A2 − 4B.
2

It is proved in the Appendix that the pure imaginary roots of equation (15)are single, that is, their multiplicity is one. If ω ≤ 1/2, then B ≤ 0 leadingto υ 21 > 0 and υ22 ≤ 0. Therefore there is only one nonzero imaginary solution,
λ = iυ 1 with υ 1 > 0. If ω > 1/2, then B > 0 implying the sign-indeterminacy

15

of A2 − 4B. However, if A > 0 and A2 > 4B, then there are two imaginary
solutions, λ = iυ 1,2 with υ 1 > υ 2 > 0 where
√√
A + A2 − 4B
A − A2 − 4Band υ 2 =
.
(18)
υ1 =
2
2
No imaginary solution exists otherwise, which implies no occurrence of stability
switch.
The region classification of the (ω, S) plane is given in Figure 3. There theflat downward-sloping curve is the locus of A = 0 below which A < 0 and above
which A > 0. The C-shaped curve is the locus of A2 − 4B = 0, in the left ofwhich A2 > 4B and in the right of which A2 < 4B.8 The plane is divided into
two subregions by the vertical line ω = 1/2. In the light-gray rectangular to theleft of the line, B < 0 due to the definition of B. The other subregion on theright is further divided into three parts by the A = 0 and A2 = 4B curves. In
the dark-gray region surrounded by the ω = 1/2 line and the upward-sloping
part of the A2 = 4B curve, A > 0, B > 0 and A2 > 4B. Thus the numberof different imaginary roots with positive imaginary parts is zero in the white
region, one in the light-gray region and two in the dark-gray region. Our first
result in this section concerns dynamics in the white region of Figure 3:
Proposition 4 Assume heterogeneous delays with ω > 1/2. If A > 0 and A2 <4B or A < 0, then the stability of the monopoly equilibrium does not switch for
any T ≥ 0.

Figure 3. Region classification of the (ω, S) plane
8 By the coefficient substitutions, we have A = 0 as

γ 2 (1 − ω)2 S 2 + 2γωS − 1 = 0
and A2 − 4B = 0 as

2 2
γ (1 − ω)2 S 2 + 1 + 2γωS − 4γω(1 + 2γωS) = 0.

In Figure 3, the curves are illustrated under γ = 1. However this specification does not affectthe relative location of the curves.

16

We verify the emergence of a limit cycle through Hopf bifurcation in the blue
and red regions. Selecting T as the bifurcation parameter, we determine thesign of the derivative of Re λ(T ) at the points where λ(T ) is purely imaginary.
Differentiating implicitly the characteristic equation (15) with respect to T , weobtain −1dλ(2λ + a1 )eλT + b2 T− .
=dTλ(b2 λ + b1 )λSubstituting λ = iυ 1,2 and arranging terms, we can arrive at the following
relationd(Re λ) = sign 2υ 21,2 − A .
sign

dT
λ=iυ1,2Inserting the expressions for υ 1 and υ 2 in (18), it can be seen that the sign ispositive for υ 1 and negative for υ 2 ,
sign
and




d(Re λ) d(Re λ) 
2 − 4B =⇒
A
>0
=
sign
dT λ=iυ1dT λ=iυ1 d(Re λ) d(Re λ) 2sign= sign − A − 4B =⇒< 0.
dT λ=iυ2dT λ=iυ2

(19)

(20)

Dynamics is qualitatively different according to whether ω is less or greater than
1/2 or more intuitively whether the parameters are selected from the blue or
the red region in Figure 3. We consider the problem of stability switches in thefollowing two cases, ω ≤ 1/2 and ω > 1/2.

4.1

Case I: ω ≤ 1/2.

In this case, the parameters are selected from the blue region in Figure 3. As isalready seen, we have only one imaginary root, λ = iυ 1 with υ 1 > 0. Inequality(19) implies that crossing of the imaginary axis is from left to right as T increases. The stability of the monopoly equilibrium can be lost and not regained.
From (16), we haveω(21)cos υ 1 T = −(1 − ω)(1 + υ 21 S 2 )andsin υ 1 T =υ 1 (1 − γωS + υ 21 S 2 ).
γ(1 − ω)(1 + υ21 S 2 )(22)

There is a unique υ1 T in the interval (0, 2π] such that υ 1 T satisfies both (21)and (22). From (21) the explicit solution of T is given by1ω
cos−1 −
.
(23)
T1 =
υ1
(1 − ω)(1 + υ 21 S 2 )17

Since1 − γωS + υ 21 S 2≥ 1 − γωS + S 2 A
2

2 
1
= 1 − γωS + S2 γ 2 (1 − ω)2 + 2γω
S − S22 2
2> 0,= S γ (1−ω)
2

we have sin υ 1 T > 0. This is the smallest stability switch, and at all stability
switches the steady state can only lose stability. Notice the following two issues:
one is that such a critical value of T is a function of S and the other is that themonopoly equilibrium is asymptotically stable when 0 < T < T1 and unstablefor T > T1 , given S. Their graphical representation with the same parametersetting except ω = 2/5 is given in Figure 4(A) in which the stability region
is colored in yellow and its boundary (23) is the partition curve in red. The
C-shaped partition curves with n = 1, 2, ..., 7 are added in order to compare thestability regions with heterogeneous weights with the one with heterogeneous
delays. The vertical dotted line S = 5 does not cross the right most C-shaped
partition curve, implying that the monopoly equilibrium is always stable if n = 1
and S = 5. The bifurcation diagram with respect to T along the S = 5 line is
illustrated in Figure 4(B) where the blue curve is the locus of the maximum
and minimum of q(t) under heterogeneous delays and the red curves are thecorresponding loci under heterogeneous weights. For graphical simplicity, welabel only TA2 and TB2 on the horizontal axis. Figures 4(A) and 4(B) capture thefollowing three facts. First of all, the C-shaped partition curves tend to the red
curve from above as n increases to infinity, implying that the stability region
under heterogeneous delays is the smallest. Secondly, due to the different shapesof the partition curves, stability switch occurs only once under heterogeneous
delays whereas it occurs twice under heterogeneous weights. And finally, thelimit cycle under the heterogeneous delays is larger than the ones under theheterogeneous weights.
Proposition 5 Assume the heterogeneous delays with ω ≤ 1/2. Then there is
only one critical value T1 for each S, and the monopoly equilibrium is asymptotically stable for T < T1 while it becomes locally unstable and bifurcates to alimit cycle for T > T1 .

18

(A) The partition curves(B) The bifurcation diagrams

Figure 4. Heterogeneous weights and delays with ω ≤ 1/2

4.2

Case II: ω > 1/2.

In this case, the parameters are selected from the red region of Figure 3. The
inequalities in (19) and (20) imply that crossing from left to right with increasing
T occurs if λ = iυ 1 and crossing from right to left occurs if λ = iυ 2 . Fromequation (16), we obtain the following two sets of the values of T for which
there are imaginary roots: for i = 1, 2 and k = 0, 1, 2, ...,
Tk,i =

θi
2kπ
+
υi
υi

(24)

where 0 ≤ θi = υi Tk,i < 2π andcos θ i = −ω
(1 − ω)(1 + υ 2i S 2 )υ i (1 − γωS + υ 2i S 2 ).
sin θi =γ(1 − ω)(1 + υ 2i S 2 )(25)

In the red region of Figure 3, we have qualitatively different dynamics fromdynamics obtained in the blue region; namely, multiple stability switches under the heterogeneous delays and an emergence of complex dynamics involving chaos. Dynamics is investigated under the same parameter setting except
ω = 4/5. We start with the case with k = 0. Both T0,1 and T0,2 obtained from(24) form the partition curve dividing the whole region of (S, T ) into stable andunstable regions. In Figure 5(A), the stability region under the heterogeneous
delays is colored in yellow and bordered by the red locus of T0,1 and the blue
locus of T0,2 while the partition curves with n = 2, 3, ..., 7 under the heterogeneous weights are depicted as black curves as in Figure 2. Given S = 5, delays
19with n = 1, 2, 3 become harmless as the dotted vertical line is located to the leftof the partition curve with n = 3. In Figure 5(B), the bifurcation diagram under
the heterogeneous delays is depicted in blue while those under the heterogeneous
weights in red. Seeing Figures 5(A) and 5(B) carefully, we have the following
observations. The boundary of the yellow region takes a C-shaped profile whichthen implies four issues concerning dynamics under the heterogeneous delays:
the first is that under the heterogeneous delays T can be harmless when S isless than S0 , the second is that stability switch occurs at least twice when S
is larger than S0 , the third is that a limit cycle emerges when the stability islost, expands, shrinks and then disappears when the stability is regained andthe fourth is that the cycle under heterogeneous delays is larger than any cycles
under heterogeneous weights.

(A) The partition curves(B) The bifurcation diagram

Figure 5. Heterogeneous weights and delays with ω > 1/2

4.3

Finite Number of Stability Switches

Returning to equation (24), we find that the partition curves under heterogeneous delays are also defined for k ≥ 1. The partition curve for k = 0 isconstructed by connecting T0,1 and T0,2 . As has been pointed out, its C-shaped
profile implies that stability switches occur twice. In the same way the Cshaped partition curve for k ≥ 1 can be constructed by connecting Tk,1 andTk,2 . Therefore there can be a finite number of switches between stability andinstability. We present numerical investigations of the multiple stability switches
in Figure 6 where the same parameter setting as in Figure 5 is used. The threshold value of S below which the positive delay of T become harmless is obtained
as S0 ≃ 4.41. To examine the existence of finite number of stability switches,
we select a particular value of S at S1 = 5.5 > S0 and then perform simulations to see how the dynamics changes when the delay T is increased from zero,
fixing S = S1 . Figure 6(A) illustrates four partition curves for k = 0, 1, 2, 3. Notice that increasing the value of parameter k shifts the corresponding partition20

curve vertically upwards. When T is increased along the vertical dotted line at
S = S1 , the number of roots with Re λ > 0 is increased by two when T passes
through a value of Tk,1 and it is decreased by two when T passes through avalue of Tk,2 for k = 0, 1, 2, 3. These critical values are denoted by the black
dots in Figure 6(A) and are in the following order,
T0,1 < T0,2 < T1,1 < T1,2 < T2,1 < T3,1 < T2,2 .
Since d(Re λ)/dT > 0 at T = Tk,1 and d(Re λ)/dT < 0 at T = Tk,2 , two roots
have positive real parts for T > T2,1 and two additional roots with positive
real parts are added when T > T3,1 . Then only two roots turn to have negative
real parts but the other two roots still have positive real parts at T = T2,2 atwhich stability switch from instablity to stablity cannot occur. In consequence,
the monopoly equilibrium remains unstable for T > T2,1 even though T passes
through the critical values of Tk,i > T2,1 . Hence in this example stability switch
occurs three times. The equilibrium is locally stable in the three intervals,[0, T0,1 ), (T0,2 , T1,1 ) and (T1,2 , T2,1 ) while it is locally unstable in the threeintervals (T0,1 , T0,2 ), (T1,1 , T1,2 ) and (T2,1 , ∞]. In Figure 6(B), we depict abifurcation diagram with respect to T and observe that the monopoly equilibrium bifurcates to a limit cycle when it loses stability and that the bifurcation
scenario is simple in the sense that only limit cycles are repeatedly born finitely
many times.
We can also prove analytically that stability switch can occur only finitely
many times. Notice that
∆1,k = Tk,1 − Tk−1,1 =

2π
υ1∆2,k = Tk,2 − Tk−1,2 =

2π
,
υ2

and

and since ∆1,k < ∆2,k , after finitely many steps a pair Tk−1,1 , Tk,1 of points
will be between two consecutive Tk,2 values. At this point two pairs of complex
roots will have positive real parts and only one of them is able to regain the21

negativity of its real part, so the monopoly equilibrium remains unstable.

(A) The partition curves(B) Multiple births of a cycle

Figure 6. Several switches between stability and instability for k = 0, 1, 2, 3
We perform further simulations to investigate the dynamics for different Svalues. The value of S is increased to S2 = 7 from S1 = 5.5. As indicated in
Figure 6(A), after losing stability at the second time, the monopoly equilibrium
does not regain stability anymore. The bifurcation diagram for 36 < T < 40
(i.e., the thick part of the vertical line S = S2 in Figure 6(A)) is illustrated
in Figure 7(A). It shows that the limit cycle becomes smaller as T increases
from 36. The cycle, however, does not merge with the equilibrium point as in
Figures 2 but bifurcates to a periodic or aperiodic cycle whose local maximum
and minimum points correspond to the plotted points spread over some regionof the output q(t). The value of S is further increased to S3 = 10 and then the
delay dynamic system (13) is numerically simulated again along the thick partof the vertical line S = S3 . The resultant dynamics is summarized in Figures
7(B) where complex dynamics is born. The bifurcation diagram illustrates thefollowing result. The limit cycle turns to be a period-2 cycle which has two
local maxima and two local minima. Around T = 48, two extrema suddenly
appear and then the limit cycles bifurcate to complex dynamics. The bifurcation
scenario is different from the well-known period-doubling route to chaos and the
one given in Figure 7(A). We can now summarize the dynamic results underheterogeneous delays:
Proposition 6 Assume heterogeneous delays with ω > 1/2. If A > 0 and A2 >4B, then (i) there are two imaginary roots with positive real parts, and thestability of the monopoly equilibrium can change a finite number of times and(ii) the unstable equilibrium bifurcates to a limit cycle or aperiodic oscillations
as T increases when S is small or large.

22

(A) S = 7(B) S = 10

Figure 7. Bifurcation diagrams

5

Concluding Remarks

We have analytically and numerically examined dynamics of a boundedly rational monopoly with two continuous delays. The general form of the characteristic
equation was derived and then attention was focused on two special cases. In
the first case in which two continuous delays have different weights, we showthat a monopoly equilibrium is destabilized and a limit cycle is born throughHopf bifurcation with increasing length of the delay. We also show that thelimit cycle expands, shrinks and then finally merges to the monopoly equilibrium with further increase. In the second case in which one delay is continuousand the other is fixed, we demonstrate the following result, in addition to thedelay effects obtained in the first case: stability and instability of the monopoly
equilibrium are alternating finitely many times and afterwords, the equilibrium
becomes unstable. Lastly it is numerically confirmed that aperiodic dynamics
can be emerged for large values of the delays.
It is of interest to note that this study expands our earlier result in Matsumoto and Szidarovszky (2012b) where monopoly dynamics under one continuous delay can generate only simple dynamics such as limit cycles. Our analysis
implies that the presence of multiple continuous delays can result in complex
dynamics involving chaos.

23

Appendix
In this appendix we show that eigenvalues λ = iυ of equation (15) are single.
For convenience, we write equation (15) as
λ2 + a1 λ + a2 + (b1 + b2 λ)e−λT = 0.

(A-1)

If a root λ = iυ is multiple, then2λ + a1 + b2 e−λT + (b1 + b2 λ)e−λT (−T ) = 0.

(A-2)

From (A-1),
e−λT = −λ2 + a1 λ + a2
b1 + b2 λand from (A-2)
 2

λ + a1 λ + a2
= 0.
2λ + a1 + (b2 − T b1 − T b2 λ) −b1 + b2 λIt can be rewritten as

T b2 λ3 +(b2 + T b1 + T a1 b2 ) λ2 +(2b1 +T a1 b1 +T a2 b2 )λ+(a1 b1 −a2 b2 +T a2 b1 ) = 0
where coefficients are
T b2 = γ(1 − ω)T,
b2 + T b1 + T a1 b2 = γ(1 − ω) 1 + 2TS ,2b1 + T a1 b1 + T a2 b2 = γ(1 − ω)a1 b1 − a2 b2 + T a2 b1 = γ(1 − ω)





T γω
2
T
S + S2 + S



γω
T γω
1
S2 − S + S2

,



.

Simplifying with γ(1 − ω) in all coefficients reduces the last equation to
1 − Sγω + T γω2S + T + ST γω
2T
3
2
λ +
λ+
= 0.
Tλ + 1 +2
SSS2

If λ = iυ, then real and imaginary parts giveυ2 =1 − Sγω + T γωS 2 + 2T S

υ2 =2S + T + ST γω
T S2

and
implying thator

1 − Sγω + T γω2S + T + ST γω
=S 2 + 2T S
T S2
2S 2 (1 + T γω) + T (4S + 2T + T Sγω) = 0

which is impossible. Therefore all pure imaginary roots are single.

24References[1] Bellman, R. and K. L. Cooke, Differential-difference Equations, Academic
Press, New York, 1956.
[2] Bischi, G-I, C. Chiarella, M. Kopel and F. Szidarovszky, Nonlinear Oligopolies:
Stability and Bifurcations, Springer-Verlag,
Berlin/Heidelberg/New York, 2010.
[3] Cushing, J., Integro-differential Equations and Delay Models in Population
Dynamics, Springer-Verlag, Berlin/Heidelberg/New York, 1977.
[4] Gandolfo, G., Economic Dynamics,
Berlin/Heidelberg/New York, 2009.

4th Edition,

Springer-Verlag,[5] Matsumoto, A. and F. Szidarovszky, "Nonlinear Delay Monopoly with
Bounded Rationality," Chaos, Solitions and Fractals, vol. 45, 507-519,
2012a.
[6] Matsumoto, A. and F. Szidarovszky, "Boundedly Rational Monopoly with
Continuously Distributed Single Time Delay," InER Discussion Paper #180, Institute of Economic Research (InER), Chuo University(http://www2.chuo-u.ac.jp/keizaiken/discuss.htm), 2012b.
[7] Naimzada, A. and G. Ricchiuti, "Complex Dynamics in a Monopoly with aRule of Thumb," Applied Mathematics and Computation, vol. 203, 921925, 2008.
[8] Okuguchi, K. and F. Szidarovszky, The Theory of Oligopoly with MultiProduct Firms, Springer-Verlag, Berlin/Heidelberg/New York, 1990.
[9] Puu, T., "The Chaotic Monopolist," Chaos, Solitons and Fractals, vol. 5,34-44, 1995.

25