Boundedly Rational Monopoly with
Continuously Distributed Single Time Delay∗
Akio Matsumoto†

Ferenc Szidarovszky‡

Abstract
In this paper we examine dynamics of a boundedly rational monopoly
with continuously distributed time delay. Constructing the gradient dynamic system where the output change is proportional to the expected
profit and the expected demand is formed based on past data with various shapes of the weighting function, three main results are analytically aswell as numerically demonstrated: (1) the stability region depends on the
shape of the weighting function and converges to the stability region of
the fixed time delay when the shape parameter goes to infinity; (2) delay
has a threshold value below which stability is preserved and above which
it is lost; (3) the equilibrium point bifurcates to a limit cycle through Hopf
bifurcation when it loses stability.

Key words: Boundedly rational, Continuously distributed time delay,
Hopf bifurcation, Limit cycle, Gradient dynamics

JEL Classification: C62, C63, D21, D42∗ The authors highly appreciate 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 paper was prepared when the second author visited the Department of Economics, Chuo University. He
appreciated its hospitality over his stay. The usual disclaimer applies.
† 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

In most traditional models, instantaneous and complete information has been
assumed. Recent research on the dynamic behavior of the economic agents,
however, emphasizes on their bounded rationality that arises when the agents
have only limited information in making their decisions. In this paper, we build adynamic monopoly model that accounts for delay information and then explore
analytically and numerically the delay effects upon local and global dynamic
behavior of a monopolist or monopolistic firm.
In real economies there is always a delay in obtaining information and implementing decisions. In the existing literature two types of delay are usually
examined: fixed time delay and continuously distributed time delay (fixed delay
and continuous delay henceforth). The former is applicable in economic situations in which an institutionally or socially determined fixed period of timedelay is present for the agents involved. The latter is appropriate for economic
situations in which different lengths of delays are distributed over the different
agents. Uncertain delays can be modeled by continuous delays, and the sametypes of models describe the situation when the firms want to react to average
past information instead of following sudden market change. So the choice ofthe type of delays has situation-dependency and results in the use of different
analytical tools. In the cases of fixed delays the dynamic equations are delaydifferential equations where the characteristic equation is a mixed polynomialexponential equation with infinitely many eigenvalues.1 The classical book by
Bellman and Cooke (1956) offers the theory of such dynamic models. Kuang(1993) gives good theoreical foundation and comprehensive summary of application in population dynamics. In economic dynamics, Howroyd and Russel(1984) construct two linear continuous time dynamic oligopoly models and examine the effect of the delay on stability. Fixed delay dynamics has been investigated in various economic frameworks ranging from micoreconomics (i.e,oligopoly dynamics) to macroeconomics (i.e., business cycle). In the cases ofcontinuous delays the dynamic equations are Volterra type integro-differential
equations. Cushing (1977) discusses the mathematical methodology dealing
with such dynamics. Invernizzi and Medio (1991) have introduced continuous
delays into mathematical economics, and this methodology is later used to examine dynamic oligopolies by Chiarella and Khomin (1996) and Chiarella andSzidarovszky (2001, 2004). Recently Matsumoto (2009) re-examined the classical Goodwin’s accelerator business cycle by replacing fixed delay in the originalmodel with continuous delay. Dynamics generated by fixed delay and continuous delay are compared in Matsumoto and Szidarovszky (2010) in which theGoodwin (2D) model, the Kaldor-Kalecki (3D) model and the Cournot oligopoly(4D) model are examined.
In our earlier paper, Matsumoto and Szidarovszky (2011), boundedly rational monopoly is examined with one and two fixed delays. A complete stability
analysis is given and it is demonstrated that in the case of locally unstable
1 A dynamic equation with fixed delays can be called a mixed difference-differential equation. However it is pointed out by Gandolfo (2009) that such terminology is somewhat dated.

2

monopoly equilibrium only simple dynamics (i.e., limit cycle) can be born when
one fixed delay is involved while complex dynamics are reached through a perioddoubling bifurcation when two fixed delays are involved. In this paper the fixed
delay is replaced with continuous delay and in addition to complete stability
analysis, the asymptotic behavior of the equilibrium with fixed and continuous
delays will be compared.
This paper is organized as follows. In the next section, we construct a
gradient dynamic model of boundedly rational monopoly. Then in Section 3, weanalytically examine local dynamics and numerically show that the continuous
delay has a threshold value at which the monopoly equilibrium loses stability.
In Section 4 we introduce a cautious expectation formation and demonstratethat stability switch occurs twice, one switch to instability from stability for asmall delay and the other switch to stability from instability for a large delay.
Concluding remarks are given in Section 5.

2

Delay Monopoly

In this section we construct a basic dynamic model of a boundedly rationalmonopoly which produces output q with marginal cost c. The price function islinearf (q) = a − bq, a, b > 0.
However, the monopoly does not know the true price function due to incomplete information. There are several ways to deal with the behavior under suchcircumstances. If the monopoly believes in a misspecified price function andchooses its decision accordingly, then a self-confirming steady state may emerge
which is different from the stationary state with full information. Or if it does
not know certain parameters of the price function, although knowing that it islinear, then the monopoly uses a local linear approximation of the price function
based on its past output data to update its estimate.2 In this study, assumingthat the monopoly knows only a finite points of the true price function, we confine attention to a situation in which the monopoly is able to estimate the rateof a profit change by using actual prices it received in the past. The estimatedrate at a value q e of output which is believed to be close to the actual output itwould select is given bydπe= a − c − 2bq e .
dq e
So the approximating gradient dynamics isq̇ = α(q)

dπe
dq e

(1)

where α(q) is an adjustment function and the dot over a variable means atime derivative. In constructing best response dynamics, global information
2 See Chapter 5 of Bischi et al. (2010) for stability/instability of economic models with
misspecified and uncertain price functions.

3

is required about the profit function, however, in applying gradient dynamics,
only local information is needed. We make the familiar assumption that theadjustment function has linear dependency on output:
Assumption 1. α(q) = αq with α > 0.
The gradient dynamics under Assumption 1 with an expected demand q ε ispresented by(2)q̇(t) = αq(t) [a − c − 2bq ε (t)]where t denotes a point of continuous time. Since q(t) = q ε (t) for all t holds at a
stationary point, equation (2) has two stationary points; a trivial point q(t) = 0
and a nontrivial point
a−c
qM =
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 expectations. In a dynamic model with continuous time scales, time delays
can be modeled with fixed delays or continuously delays. Matsumoto and Szidarovszky (2011) examined dynamic monopoly with single and two fixed delays.
In this study we adopt a single continuous delay and consider the delay effects
on the dynamics.3 Before proceeding, we briefly summarize the results obtainedin the dynamic monopoly with one fixed delay.
We assume q e (t) = q(t − τ ) where τ > 0 denotes a fixed delay. Substitutingeq (t) for q(t − τ ) gives a nonlinear delay differential equation,q̇(t) = αq(t) [a − c − 2bq(t − τ )] .

(3)

Linearizing equation (3) and introducing the new variable, x(t) = q(t) − q M
yield the following linearized form,ẋ(t) = −γx(t − τ ) with γ = α(a − c) > 0.

Substituting the exponential solutions x(t) = x0 eλt into the linearized equation
gives the characteristic equation
λ + γe−λτ = 0.

(4)

The sufficient condition for local asymptotic stability is that the real parts ofthe eigenvalues are negative. It is shown in Matsumoto and Szidarovszky (2011)that the monopoly equilibrium is locally asymptotically stable for 0 < τ < τ ∗ ,locally unstable for τ > τ ∗ and undergoes a Hopf bifurcation at τ = τ ∗ where athreshold value τ ∗ of delay is defined byπτ∗ =.
2α(a − c)This partition curve, which divides the parameter space into a stable and unstable regions, is downward sloping with respect to γ = α(a − c). The monopoly
equilibrium is locally stable below the partition curve, locally asymptoticallyunstable above and bifurcates to a limit cycle when it crosses the curve.
3 Monopoly dynamics with two continuous delays is considered in the subsequent paper.

43Single Time Delay I

As it was mentioned earlier, continuous delay is an alternative approach to deal
with delays. The gradient dynamics with a single continuous delay is given by
the following two equations:q̇(t) = αq(t) [a − c − 2bq ε (t)]εq (t) =t
0

(5)
W (t − s, τ , m)q(s)ds

where α is the speed of adjustment and the weighting function is defined by 1 t−s− τif m = 0, τe
W (t − s, τ , m) =
(6)
 


 1 m m+1 (t − s)m e− m(t−s)τ
if m ≥ 1m! τ

where m is a nonnegative integer and τ is a positive real parameter, which is
associated with the length of the delay. The first equation of (5) implies thatthe growth rate of output is proportional to the expected demand. The second
equation indicates that the expected output at time t is the weighted averageof the actual demand in the past. According to (6), the shape of the weighting
function is determined by the value of the shape parameter, m. For m = 0,
weights are exponentially declining with the most weight given to the mostcurrent data. For m ≥ 1, zero weight is given to the most current data, rising
to maximum at s = t − τ and declining exponentially thereafter. The weights
take a bell-shaped form which becomes taller and thinner as m increases.
To consider local dynamics of this system in a neighborhood of the equilibrium point, we need to construct a linearized version. If the deviations of
the actual and expected outputs from the equilibrium value are denoted byxδ (t) = q(t) − q M and xεδ (t) = q ε (t) − q M , then the linearized system with
continuous delay can be written asẋδ (t) = −γxεδ (t),xεδ (t) =

 t
0

(7)
W (t − s, τ , m)xδ (s)ds,

where γ = 2αbq M . To examine dynamics of system (7), we substitute thesecond equation of (7) into the first to obtain the following Volterra-type integrodifferential equation:
 tẋδ (t) + γ
W (t − s, τ , m)xδ (s)ds = 0.
0

5

Looking for the solution in the usual exponential form xδ (t) = x0 eλt and substituting it into the above equation, we obtain t
λ+γ
W (t − s, τ , m)e−λ(t−s) ds = 0.
0Introducing the new variable z = t − s simplifies the integral as t t
W (t − s, τ , m)e−λ(t−s) ds =W (z, τ , m)e−λz dz.
00

By letting t → ∞ and assuming that Re(λ) + m
τ > 0, we have ∞1 − z −λze τedz = (1 + λτ )−1 if m = 0
τ
0
and
 ∞
0

That is,1  m m+1 m − mz
λτ
−λz
τ
z e
e
dz = 1 +m! τm−(m+1)

λτdz = 1 +m̄

−(m+1) ∞−λz

W (z, τ , m)e

0

with

m̄ =


 1


if m ≥ 1.

if m = 0,m if m ≥ 1.

Then the characteristic equation becomesτ m+1+ γ = 0.
λ 1+ λm̄(8)Expanding the characteristic equation presents the (m + 2)-th order polynomial
equationa0 λm+2 + a1 λm+1 + ... + am+1 λ + am+2 = 0
where the coefficients ai are given as τ m+1−k  m + 1
ak =
k
m̄

for 0 ≤ k ≤ m,

am+1 = 1 and am+2 = γ.
In the case of the high order polynomial equation, the Routh-Hurwitz theorem4
provides the necessary and sufficient conditions for all the roots to have negative
4 See, for example, Gandolfo (2009) for this theorem.

6

real parts. Applying the theorem, we first need to construct the Routh-Hurwitz
determinant:0
a1 a0 0 0 · · · a3 a2 a1 a0 · · ·0  a5 a4 a3 a2 · · ·0 .
Dm+2 = det  a7 a6 a5 a4 · · ·
0 


 ·
·
·
· ···
0 
0 0 0 00 am+2
Then the stability conditions are as follows:(1) all coefficients are positive, ak > 0 for k = 0, 1, 2, ..., m + 2,(2) the principal minors of the Routh-Hurwitz determinant are all positive,
m+1
2
3
Dm+2
> 0, Dm+2> 0, ..., Dm+2>0kis the k-th order leading principal minor of Dm+2 . Noticewhere Dm+2
m+1
m+2
that Dm+2 > 0 always leads to Dm+2
> 0 since am+2 = γ > 0.

Since it is difficult to obtain a general solution of equation (8), we draw
attention to special cases with m = 0, 1, 2, 3 and m → ∞ and examine stabilityof the monopoly equilibrium analytically as well as numerically.
Case I-0. m = 0.
Substituting m = 0 reveals that the characteristic equation (8) is quadratic,
τ λ2 + λ + γ = 0 where all coefficients are positive. It does not have nonnegativeroots and the real parts of the complex roots are negative. Thus the monopoly
equilibrium is locally asymptotically stable for all τ > 0. Since the delay does
not affect asymptotic behavior of the monopoly equilibrium, such a delay iscalled harmless.
Case I-1. m = 1.
The characteristic equation (8) with m = 1 becomes cubic and its coefficientsare all positivea0 = τ 2 > 0, a1 = 2τ > 0, a2 = 1 > 0, a3 = γ > 0.
According to the Routh-Hurwitz criterion, the following leading minor of D3
needs to be positive for preserving stability of the equilibrium, a a0 
.
D32 =  1
a3 a2 To obtain D32 = τ (2 − τ γ) > 0, the delay τ should be less than the threshold
value
2
(9)
τ ∗1 = .
γ7There is a possibility of the emergence of a limit cycle when loss of stability occurs at τ = τ ∗1 . The Hopf bifurcation theorem comes in to provide the sufficientconditions for it:(H1) the characteristic equation of the dynamic system has a pair of pure
imaginary roots and has no other roots with zero real parts;(H2) the real parts of these roots vary with a bifurcation parameter.
The D32 = 0 curve divides the parameter space into stable and unstableparts. Substituting a3 = a1 a2 /a0 into the characteristic equation gives thefactored form
(a1 + a0 λ)(a2 + a0 λ2 ) = 0.
We have therefore three characteristic roots, two purely imaginary roots andone real and negative root,

a2
1
a1
2
λ1,2 = ± − = ±i and λ3 = − = − < 0.
a0
τ
a0
τ
The first condition (H1) of the Hopf theorem is satisfied.
Next we select the delay τ as the bifurcation parameter and consider theroots of the characteristic equation as continuous functions of τ :τ 2 λ(τ )3 + 2τ λ(τ )2 + λ(τ ) + γ = 0
Differentiating it with respect to τ gives2τ λ3 + 2λ2
dλ=− 2 2.
dτ
3τ λ + 4τ λ + 1
Substituting λ = i/τ , rationalizing the right hand side and noticing that theterms with λ and λ3 are imaginary while the constant and the term λ2 are real
yield the following form of the real part of the derivative of λ with respect to τ :

 
1
dλ 
= 2 > 0.
Redτ λ= i5ττThe last inequality indicates that the second condition (H2) is also satisfied. The
real parts of the complex roots change to positive from negative value resulting in
the loss of stability on the partition curve. Hence the Hopf bifurcation theorem
confirms the birth of a limit cycle when stability is lost.
We numerically examine the analytical result just obtained. The dynamic
system under the investigation is obtained by substituting m = 1 into (5),q̇(t) = αq(t) [a − c − 2bq ε (t)]εq (t) =

t 
0

1
τ

2
t−s

(t − s)e− τ q(s)ds.
8

Differentiating the second equation with respect to t and introducing a newvariable t
1 − t−sq0 (t) =e τ q(s)ds
0 τtransforms the dynamic system with continuous delay into a 3D system of ordinary differential equationsq̇(t) = αq(t) [a − c − 2bq ε (t)] ,q̇ ε (t) =1(q0 (t) − q ε (t)) ,τq̇0 (t) =1(q(t) − q0 (t)) .
τWe specify the parameters’ values as a = 2, b = 1, c = 1, α = 1,5 and take
τ = 3 and the initial values of all variables to be q M − 0.1. Then simulating the
3D system exhibits the birth of a limit cycle as illustrated in Figure 1 where a
black trajectory starting at the black dot (i.e., positive initial point) converges
to a red cycle in the (q, q ε , q0 ) space. We then summarize this result as follows:
the monopoly equilibrium point with m = 1 is destabilized through a Hopf
bifurcation and, as it is numerically confirmed, converges to a limit cycle when
the delay τ is larger than the critical value 2/γ.

q 0 Ht L

qHtL
e

q HtL
Figure 1. Convergence to a limit cycle in the (q, q ε , q0 ) plane
Case I-2. m = 2.
The characteristic equation (8) with m = 2 becomes quartic
a0 λ4 + a1 λ3 + a2 λ2 + a3 λ + a4 = 0
5 This set of the parameters are repeatedly used in the following numerical examples. Noticethat γ = 1 under this set.

9

and its coefficients are all positive,
a0 = τ 3 > 0, a1 = 6τ 2 > 0, a2 = 12τ > 0, a3 = 8 > 0, a4 = 8γ > 0.
The principal minors of the Routh-Hurwitz determinant are D42 = 64τ 3 > 0 and
D43 = 32τ 3 (16 − 9γτ ). To obtain D32 > 0, the delay τ should be less than thethreshold value
16
1.78
≃
.
(10)
τ ∗2 =
9γγ
Thus the equilibrium is locally asymptotically stable if τ < τ ∗2 and locallyunstable if τ > τ ∗2 . In the same way as in Case I-1, we can show the existenceof a limit cycle at the critical value τ ∗2 . In particular, solving D43 = a1 a2 a3 −(a0 a23 + a21 a4 ) = 0 for a4 , substituting it into the characteristic equation andfactoring the resultant equation yield(a3 + a1 λ2 )(a1 a2 − a0 a3 + a21 λ + a0 a1 λ2 ) = 0.
The solutions of a3 + a1 λ2 = 0 are purely imaginary,2
λ1,2 = ±i √ ,τ 3and the other two characteristic roots are the solutions of the quadratic equation(a1 a2 − a0 a3 ) + a21 λ + a0 a1 λ2 = 0,√
−9 ± i 15
λ3,4 =
3τ
whose real parts are negative. The first condition (H1) of the Hopf bifurcation
theorem is satisfied.
To confirm the second condition, we choose τ as the bifurcation parameter
again and differentiate the characteristic equation with respect to τ to have3τ 2 λ4 + 12τ λ3 + 12λ2
dλ=− 3 3.
dτ
4τ λ + 18τ 2 λ2 + 24τ λ + 8
2and taking the real part, we have
Substituting λ = i τ √
3





0.316
dλ 
6
≃
> 0.
Re
=
dτ λ=i √2
19τ 2
τ2
τ

3

We thereby confirm that the monopoly equilibrium with m = 2 is destabilized
through a Hopf bifurcation when the delay τ crosses the critical value τ ∗2 .

Case I-3. m = 3.

10

The characteristic equation (8) is qintic and its coefficients are all positivea0 = τ 4 > 0, a1 = 12τ 3 > 0, a2 = 54τ 2 > 0, a3 = 108τ > 0, a4 = 81 > 0, a5 = 81γ > 0.
The first two principal minors of the Routh-Hurwitz determinant are positive
D52 = 540τ 5 > 0 and D53 = 972τ 6 (48 + γτ ) > 0.
The sign of the fourth order principal minor D54 = −6561τ 6 (γ 2 τ 2 +336γτ −576)
is ambiguous. To obtain D54 > 0, the delay τ should be less than the threshold
value
√
1.71
24(5 2 − 7)
τ ∗3 =≃.
(11)γγ
Although we omit the detail, we can also show that the continuous delay system(5) with m = 3 can generate a limit cycle through a Hopf bifurcation in the same
way as in the previous cases when the monopoly equilibrium loses stability.
The relations (9), (10) and (11) 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 the stability (yellow) region with fixed delay defined byτ γ < 2 are depicted in Figure 2. It can be seen that all curves are hyperbolicand are approaching the red-colored boundary of the stability region from above.
In other words, the stable region with continuous delay becomes smaller as thevalue of m increases and converges to the region defined by the fixed delay whenm tends to infinity. This result obtained are natural if we notice the propertiesof the weighting function. The weighting function for m ≥ 1 is a bell-shaped
and becomes more peaked around t − s as m increases. Furthermore it tends to
the Dirac delta function if m → ∞. In consequence, for sufficiently large m, theweighting function may be regarded as very close to the Dirac delta function andthe dynamic behavior under the continuous delay is very similar to that under
the fixed delay. We can explain this phenomenon mathematically by noticingthat the characteristic equation (8) of the continuously distributed case can be
written as−(m+1) 
−1
τλ
τλ
= 0,
1+
λ+γ 1+mm
and as m → ∞, the left hand side converges to
λ + γe−λτ = 0.

This is the characteristic equation of the delay differential equation with a single
fixed delay and is identical with equation (4). In short, under continuous delay,
although we comprehensively use all delayed or past output data, the stability
domain is sensitive to the shape of the weighting function. Hence we obtain thefollowing two results:
Proposition 1 (1) The monopoly equilibrium of the continuously distributed
single delay model is always stable for any delays if m = 0 and is destabilized
through a Hopf bifurcation if m ≥ 1; (2) The stability region decreases as mincreases (i.e., τ ∗ < τ ∗3 < τ ∗2 < τ ∗1 ) and converges to the stability region obtained
under the fixed delay when m goes to infinity (i.e., τ ∗m → τ ∗ as m → ∞).
11

Figure 2. Three partiton curves and stability region

4Single Time Delay II

In this section, we introduce a cautious expectation formation defined byq ε (t) = ωq e (t) + (1 − ω)q(t)eq (t) =t
0

(12)
W (t − s, τ , m)q(s)ds

with 0 < ω ≤ 1. The expectation is formed with two steps: the weighted averageof the past data is calculated at the first step and then the expected demandis selected, at the second step, somewhere in between the current output level
and the weighted average level.6 If m ≥ 1, then zero weight is given to themost current data in the weighting function, and so the second term in the firstequation of (12) gives a larger weight to it taking a certain learning procedure
based on past data at the second stage. We examine in some detail the dynamic
effects caused by a single delay with the cautious expectation formation. Following the method we take in the previous section, the characteristic equationof the system (12) can be obtained as




m+1
m+1
λτ
λτ
=0
(13)
λ 1+
+ γ ω + (1 − ω) 1 +m̄
m̄
6 Since the first equation of (12) can be rewritten asq ε (t) − q(t) = ω(q e (t) − −q(t)).
It can be mentioned that the expected demand is formed in such a way that the expectation
error is proportional to the difference between the weighted average level and the current level.

12

which is reduced to equation (8) if ω = 1. To find how the shape of the weighting
function, W (t − s, τ , m), affects the dynamics of q M , we sequentially increase
the value of m from zero to five and then to infinity.

Case II-0 m = 0
Substituting m = 0 in equation (13) presents the form
λ(1 + λτ ) + γω + γ(1 − ω)(1 + λτ ) = 0orτ λ2 + (1 + τ γ(1 − ω))λ + γ = 0.

Since all coefficients are positive, there is no nonnegative root and the real partsof the complex eigenvalues are negative,Re(λ± ) = −

1 + (1 − ω)τ γ< 0,2τ

implying that the equilibrium is locally asymptotically stable for all τ > 0. As
in Case I-0, the coninuous delay is again harmless when the weight exponentiallydeclines (i.e., m = 0).
For m ≥ 1, expanding the characteristic equation (13) yields the polynomial
equation of degree m + 2
b0 λm+2 + b1 λm+1 + · · · + bm λ2 + bm+1 λ + bm+2 = 0(14)where the coefficients are defined by
b0 = a0 ,bk = ak + γ(1 − ω)ak−1 for 1 ≤ k ≤ m,
bm+1 = am+1 + γ(1 − ω)am ,bm+2 = γwithak = τ m+1−k  m + 1
mkCase II-1 m = 1

13

for 0 ≤ k ≤ m + 1.

(15)

Equation (13) with m = 1 becomes cubic:b0 λ3 + b1 λ2 + b2 λ + b3 = 0(16)

where the coefficients are defined by
b0 = τ 2 , b1 = 2τ + γ(1 − ω)τ 2 , b2 = 1 + 2τ γ(1 − ω) and b3 = γ.
All coefficients are positive, so the Routh-Hurwitz criterion implies that themonopoly equilibrium is locally asymptotically stable if

b1 b0
2
>0
(17)
D3 = detb3 b2
where the determinant is written asτ 2γ 2 (1 − ω)2 τ 2 + (4γ − 5γω)τ + 2 .
Since τ > 0, this expression is positive if and only if

f (τ γ) = 2(1 − ω)2 (τ γ)2 + (4 − 5ω)τ γ + 2 > 0.

(18)f(τ γ) is quadratic with respect to τ γ and its discriminant has the form
D = ω(9ω − 8).
If ω < 8/9, then D < 0 so (18) always holds. If ω = 8/9, then the right handside of (18) simplifies as
2
τγ
−1
f(τ γ) = 2
9so (18) holds if τ γ = 9. If ω > 8/9, then D > 0 and f (τ γ) has two distinct realroots:5ω − 4 − ω(9ω − 8)5ω − 4 + ω(9ω − 8)
τ γA =
and
τ
γ
=
.
B
4(1 − ω)24(1 − ω)2
f(0) > 0 and f ′ (0) < 0 imply that both roots are positive. So the monopoly
equilibrium is locally asymptotically stable if ω > 8/9 and τ γ < τ γ A or τ γ >τ γB .
These analytical results are graphically visualized in Figure 3. The locus ofb1 b2 −b3 b0 = 0 is depicted as the blue-red curve passing through points A, C and
B and partitions the nonnegative (ω, τ γ) plane into a stable (white) region anda unstable (orange) region. The monopoly equilibrium is locally asymptoticallystable regardless of the value of τ γ if ω < 8/9. Stability-switch occurs twice if
8/9 < ω < 1. The vertical real line at ω = ω̄ > 8/9 crosses the partition curveat two points A and B whose ordinates are τ γ A and τ γ B , respectively. The
monopoly equilibrium is locally stable for τ = 0, loses stability at point A andgains stability at point B when τ γ increases from zero along the vertical line atω = ω̄. For ω = 1, the stability condition (18) is reduced to−τ γ + 2 > 0.
14

The monopoly equilibrium is locally stable if τ γ < 2 and locally unstable if
τ γ > 2, implying that the stability switch occurs only once at τ γ = 2 forω = 1, which is the same as the result obtained in Case I-1. This coincidence isreasonable since Case I-1 is identical with Case II-1 for ω = 1.

Figure 3. Stability region with m = 1
The local behavior of the equilibrium at points A and B has been analyticallyexamined. However, global dynamic behavior of the locally unstable equilibrium
between points A and B is still in question. To investigate such behavior, we willshow that at these critical values Hopf bifurcation occurs giving the possibilityof the birth of limit cycles even under the cautious expectation formation.
We start with the first condition, (H1). The cubic characteristic equation(16) can be factored when b1 b2 − b0 b3 = 0,
(b1 + b0 λ)(b2 + b0 λ2 ) = 0.
This factorization implies that there are a pair of purely imaginary roots andone negative root. The two purely imaginary roots are given byb2
λ1,2 = ± − = ±iβb0
withand the negative root by
1 + 2τ γ(1 − w)β=τλ3 = −b1
2 + (1 − w)τ γ
.
=−
b0
τ

The fulfilment of (H1) is confirmed.

15

We turn to verification of (H2). Selecting τ as the bifurcation parameter wemight treat the eigenvalue as a continuous function of τ , λ = λ(τ ). Differentiating the characteristic equation (16) implicitly with respect to τ and arranging
terms, we havedλ2λ3 τ + λ2 (2 + 2τ γ(1 − ω)) + 2λγ(1 − ω).
=− 2 2dτ
3λ τ + 2λ(2τ + γ(1 − ω)τ 2 ) + (1 + 2τ γ(1 − ω))(19)At λ = iβ,dλ(2τ β 3 − 2βγ(1 − ω))i + β 2 (2 + 2τ γ(1 − ω))=dτ−2β 2 τ 2 + 2βi(2τ + γ(1 − ω)τ 2 )where the relation (βτ )2 = 1 + 2τ γ(1 − ω) is used to simplify the denominatorof the above equation. Then the real part becomes 
1 − γ 2 τ 2 (1 − ω)2
dλ 
= 2 4
Re

dτ λ=iββ τ + (2τ + γ(1 − ω)τ 2 )2with the positive denominator. It is easy to see thatτ γA <so1< τ γB ,
1−ω


 


dλ 
dλ 
> 0 at point A and Re< 0 at point B.
Redτ λ=iβdτ λ=iβ

So at point A, the real part changes sign from negative to positive and at point
B, from positive to negative. This demonstrates that (H2) of the Hopf theorem
is satisfied. So Hopf bifurcation occurs at both points.
We next numerically examine the switching of stability and global behavior.
The dynamic system under the investigation (i.e. m = 1) is obtained as,q̇(t) = αq(t) [a − c − 2b (ωq ε (t) + (1 − ω)q(t))]ε

q (t) =

t
0

1
t−s
(t − s)e− τ q(s)ds.
τ2

(20)

Differentiating the second equation with respect to t and introducing a newvariable t
1 − t−se τ q(s)ds
q0 (t) =τ
0

16

transform the dynamic system with continuously distributed time delay into a
3D system of the ordinary differential equationsq̇(t) = αq(t) [a − c − 2b (ωq ε (t) + (1 − ω)q(t))] ,q̇ ε (t) =1(q0 (t) − q ε (t)) ,τq̇0 (t) =1(q(t) − q0 (t)) .
τ(21)

We use the same parameter set (i.e., a = 2, b = 1, c = 1, α = 1) and thesame initial values (i.e., q(0) = q ε (0) = q0 (t) = q M − 0.1) as in Case I-1.
Increasing the value of τ γ from τ γ A to τ γ B along the vertical line ω̄ = 0.91 in
Figure 3, we obtain the bifurcation diagram illustrated in Figure 4 where thelocal maximum and minimum are plotted against each value of τ γ. It can beseen that the monopoly equilibrium loses stability bifurcating to a limit cycle
when τ γ increases to τ γ A and gains stability when τ γ arrives at τ γ B . It is also
observed that the amplitude of the limit cycle first increases and then decreases
as τ γ increases from τ γ A to τ γ B .

Figure 4. Bifurcation diagram along the ω = 0.91 line

We summarize the results obtained in Case II-1 as follows:
Proposition 2 Under the cautious expectation formation, dynamics of the monopoly
equilibrium q M with τ > 0, m = 1 and 0 < ω ≤ 1 takes one of the following
alternative behavior;(I) q M is locally asymptotically stable if 0 < ω < 8/9 regardless of the valuesof τ γ (i.e., the delay is harmless);
17

(II) q M bifurcates to a limit cycle at τ γ A (ω) and the limit cycle de-bifurcatesto the monopoly equilibrium at τ γ B (ω) if 8/9 < ω < 1 where τ γ A (ω) andτ γ B (ω) are the bifurcation values depending on ω (i.e., stability switch
occurs twice);(III) q M bifurcates to a limit cycle at τ γ = 2 and never regains stability forτ γ > 2 if ω = 1 (i.e., stability switch occurs once).

Case II-2 m ≥ 2
As in the same way as in Case II-1, we can check the stability condition, thebirth of a limit cycle and stability switch in the case of m ≥ 2. For example, thecharacteristic equation (13) with m = 2 is quartic,b0 λ4 + b1 λ3 + b2 λ2 + b3 λ + b4 = 0
and the stability conditions are given by
D42 = det



b1
b3

b0
b2



b1
> 0 and D43 =  b3
0

b0
b2
b4

where the coefficients are given by
b0 =


0
b1  > 0b3

τ33τ 2 τ 3 γ(1 − ω), b1 =
+
,
8
4
8

3τ
3
3τ 2 γ(1 − ω)+, b3 = 1 + τ γ(1 − ω), b4 = γ.
242
3
Since all coefficients are positive and D4 > 0 implies D42 > 0, it remains to check
whether D43 can be positive. Notice thatb2 =D43 = α0 (τ γ)3 + α1 (τ λ)2 + α2 (τ γ) + α3
where

(22)

α0 = (9ω − 8)(1 − ω)2α1 = −12(4 − 9ω + 5ω 2 )α2 = −12(8 − 11ω)α3 = −64

The discriminant of the cubic equation (22) has the form
∆ = 442368(1 − ω)3 ω 2 (5ω − 4)
18

(23)

which is obtained by substituting αi of (23) into the definition of the discriminant, −4α0 α32 + α21 α22 − 4α31 α3 + 18α0 α1 α2 α3 − 27α20 α23 . If ω < 4/5, then ∆ < 0
implying D43 > 0 where (23) has a pair of conjugate complex roots and one negative real root. If ω = 4/5, then ∆ = 0 implying that D43 = (10−τ γ)2 (20+τ γ) > 0
for τ γ = 10 where (23) has equal roots at the red point on the D43 = 0 locus. If ω > 4/5, then ∆ > 0 implying that D43 = 0 has three distinct realroots. Since equation (22) is cubic, it is possible to derive explicit forms of thereal roots. However, to simplify the analysis, we numerically obtain the roots.
Taking ω 0 = 0.815 in addition to the parametric set, we have two real roots,τ γ A ≃ 5.42, τ γ B ≃ 23.75 and one negative root. As seen in Figure 5, D43 ≤ 0 forτ γ ∈ [τ γ A , τ γ B ] which is an unstable interval and D43 > 0 for 0 ≤ τ γ < τ γ A or
τ γ > τ γ B . Stability switch occurs twice at τ γ = τ γ A and τ γ = τ γ B . Further
the cubic equation is reduced to a quadratic equation for ω = 8/9 implying thatthe locus of D43 = 0 is defined only for ω < 8/9 and asymptotic to the verticalline at ω = 8/9. If 8/9 < ω < 1, then the cubic equation (22) has one positiveroot τ γ C for ω = ω1 . It is confirmed that D43 > 0 for τ γ < τ γ C and D43 ≤ 0otherwise. The equilibrium point switches to be unstable at point C on theD43 = 0 locus when the delay is increased along the vertical line at ω = ω 1 .
The dark gray region is the unstable region for m = 1 and adding the light grayregion to it gives the unstable region for m = 2.
We turn now to show that Hopf bifurcation can occur on the curve D43 = 0.
When D43 = 0 holds, the characteristic equation is factored as(b3 + b1 λ2 )(b1 b2 − b0 b3 + b21 λ + b0 b1 λ2 ) = 0.
It is clear that the characteristic equation has a pair of purely imaginary roots,b3
λ1,2 = ± − = ±iβ
b1
with
β=



4(2 + 3γ(1 − ω)τ )6τ 2 + γ(1 − ω)τ 3

and the real parts of other two roots are not zero. So condition (H1) is satisfied.
Assuming that the characteristic root depends on τ and then differentiating
the characteristic equation with respect to τ , we obtain the derivative:
3γ(1−ω)τ 23γ(1−ω)τ
3
3τ 2 4
3τ
3
λ
+
+
+
+
λ
λ
λ2 + 3γ(1−ω)
8
2
8
2
2
2
dλ




.
=− 3
3
2
2
τ
dτ
λ3 + 3 3τ + γ(1−ω)τ λ2 + 2 3τ + 3γ(1−ω)τ λ + 3γ(1−ω) + 1
2

4

8

2

4

2

To determine stability switch, we evaluate the derivative at the purely imaginary
solution, λ = iβ. Thus
 

dλ 
3(2 − (1 − ω)τ γ)
Re
.
=
2

2
dτ λ=iβ4 + τ (16β + 9γ 2 (1 − ω)2 + 12γ(1 − ω)τ )19

We numerically confirm that the real part of the derivative is positive at pointτ γ A and negative at point τ γ B . So he second condition (H2) is also satisfied.
Therefore Hopf bifurcation occurs at both critical points.

Figure 5. The partition curves with m = 1 and m = 2
In the same way, it can be numerically confirmed that the stability condition
m+1
m+1> 0 for 2 ≤ m ≤ 5. In Figure 6, five partition curves Dm+2=0
is given by Dm+2
for m = 1, 2, 3, 4, 5 are depicted. The right most curve is the locus of D32 = 0(i.e., m = 1) and the left most curve is the locus of D76 = 0 (i.e., m = 5). The
partition curve shifts leftward with the increasing value of m. At the red dot on
m+1
each curve, equation Dm+2
= 0 has two real and equal roots.7 As is shown inCase II-1, for any m ≥ 2, stability switch occurs twice for ω greater than theabscissa of the red point while the delay becomes harmless for smaller values ofω.
Case II-3 m → ∞As m increases, the weighting function becomes more peaked around t − sand tends to the Dirac delta function. Notice that as m goes to infinity, thefirst equation of (20) is reduced to a delay differential equation,q̇(t) = αq(t) [a − c − 2b (ωq(t − τ ) + (1 − ω)q(t))](24)

and the characteristic equation (13) converges to
λ + γ(1 − ω) + γωe−λτ = 0
7 The curves look like steeply-shaped hypabolas. However, if we enlarge each curve in theneighborhood of its red point, then it can be found that the curves take the distorted C-shaped
profiles as the curve in Figure 3 or Figure 5.

20which is the characteristic equation of the delay differential equation (24). To
verify the possibility of stability switch for which the characteristic equationmust have a pair of purely imaginary conjugate roots, we can assume without
loss of generality that λ = iυ, υ > 0. By the real and imaginary parts, thecharacteristic equation is divided into two equationsγ(1 − ω) + γω cos υτ = 0,(25)

υ − γω sin υτ = 0.
By moving γ(1 − ω) and υ to the right hand side of equations in (25), squaring
and adding them together, we obtainυ 2 = γ 2 (2ω − 1)which is defined only for ω > 1/2, otherwise no stability switch occurs. We thenthink of the roots as continuous functions in terms of τ and then differentiate
the characteristic equation with respect to τ to obtain

−1
τ
dλ
eτ λ
γω
− and eτ λ = −
.
=
dτ
γωλ λ
λ + γ(1 − ω)Thusd(Re λ) dτ λ=iυ

= Re
=




dλ 
dτ λ=iυ−11

υ 2 + γ 2 (1 − ω)2>0

The last inequality implies that all the roots that cross the imaginary axis at iυcross from left to right as τ increases.
From (25), we haveγω cos υτ = −γ(1 − ω),(26)γω sin υτ = υ.
Hence there is a unique υτ , π/2 < υτ < π such that υτ makes both equationsin (26) hold. Using the first equation we derive the partition curve

1−ω
−1
cos
−
ω
√
γτ =
2ω − 1
which is defined for ω > 1/2. In Figure 6, in addition to the five partition curves,the downward sloping hyperbolic red partition curve of the fixed delay case areillustrated. The monopoly equilibrium with the fixed delay is locally asymptotically stable in the yellow region. We can summarize three results obtained in
Cases II-2 and II-3:
21

Proposition 3 In the case with τ > 0 and m ≥ 1, (1) increasing m has adestabilizing effect in the sense that it decreases the stability region; (2) thestability region with continuously distributed time delay is larger than the one
with fixed time delay and the former converges to the latter as m goes to infinity;(3) the stability switch, if possible, occurs twice, implying that the equilibrium
is locally stable for smaller or larger values of τ γ while it bifurcates to a limit
cycle for medium values.

Figure 6. Stability region and five partition cuves

5

Concluding Remarks

In this paper a boundedly rational monopoly with a continuously distributed
time delay is examined. Constructing a gradient dynamic system where the rateof the output change is proportional to the derivative of the expected profit,
three main results are analytically and numerically demonstrated. First, thestability region depends on the shape parameter of the weighting function of
the past data: stability is preserved if the weights exponentially decline while
it can be lost if the weighting function takes a bell-shaped profile. Further notethat the stability region of the continuously distributed time delay converges
to the stability region of the fixed time delay when the shape parameter goesto infinity (in other word, the weighting function converges to the Dirac delta
function). Second, the stability switches indicate that the delay has a destabilizing effect. Specifically the stable equilibrium point becomes unstable andnever regains stability if the expected demand is formed based only on past data.
However, under the cautious expectation formation where the expected demand
is a weighted average of the realized output and the past data, switches to instability from stability and then to stability from instability are both possible,
depending on the value of the delay. Finally the equilibrium point bifurcates to22

a limit cycle through Hopf bifurcation when it loses stability.

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, Berlin/Heidelberg/New York,
Springer-Verlag, 2010.
[3] Chiarella, C. and A. Khomin, "An Analysis of the Complex Behavior of
Nonlinear Oligopoly Models with Time Delays," Chaos, Solitons andFractals, 7(12), 1996, 2049-2065.
[4] Chiarella, C. and F. Szidarovszky, The Birth of Limit Cycles in Nonlinear Oligopolies with Continuously Distributed Information Lags, In M.
Dror, P. L’Ecuyer and F. Szidarovszky (eds), Modeling Uncertainty,
249-268, Dordrecht, Kluwer, 2001.
[5] Chiarella, C. and F. Szidarovszky, "Dynamic Oligopolies without Full Information and with Continuously Distributed Time Lags," Journal of
Economic Behavior and Organization, 54 (4), 2004, 495-511.
[6] Cushing, J., Integro-differential Equations and Delay Models in Population
Dynamics, Springer-Verlag, Berlin/Heidelberg/New York, 1977.
[7] Gandolfo,
G.,
Economic Dynamics,
4th Edition,
Springer,
Berlin/Heidelberg/New York, 2009.
[8] Howroyd, T. and A. Russel, "Cournot Oligopoly Models with Time Delays," Journal of Mathematical Economics, 13, 1984, 97-103.
[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., Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.
[11] Matsumoto, A., "Note on Goodwin’s 1951 Nonlinear Accelerator Modelwith an Investment Delay," Journal of Economic Dynamics and Control, 33(4), 2009, 832-842.
[12] Matsumoto, A. and F. Szidarovszky, "Nonlinear Delay Monopoly
with Bounded Rationality," Discussion Paper #166, Instituteof Economic Research, Chuo University (http://www2.chuou.ac.jp/keizaiken/diccuss.htm), 2011. To appear in Chaos, Solitons andFractals.
[13] Matsumoto, A. and F. Szidarovszky, "Delay Differential Nonlinear Economic Models," in G. I. Bischi, C. Chiarella and L, Gardini (eds.) Nonlinear Dynamics in Economics, Finance and Social Sciences, SpringerVerlag, Berlin/Heidelberg/New York, 2010, 195-214.
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