NONLINEAR DELAY MONOPOLY WITH BOUNDEDRATIONALITYAKIO MATSUMOTO AND FERENC SZIDAROVSZKY

Abstract. The purpose of this paper is to study dynamics of a monopolistic…rm in a continuous-time framework. The …rm is assumed to be boundedly
rational and to experience time delays in obtaining and implementing information on output. A dynamic adjustment process is based on the gradientof the expected pro…t. The paper is divided into three parts: We examinedelay e¤ects on dynamics caused by one-time delay and two-time delays in the…rst two parts. Global dynamics and analytical results on local dynamics are
numerically con…rmed in the third part. Four main results are demonstrated.
First, the stability switch from stability to instability occurs only once in thecase of the one-time delay. Second, the alternation of stability and instability
can continue if two time delays are involved. Third, the birth of Hopf bifurcation is analytically shown if stability is lost. Finally, in a bifurcation process,there are a period-doubling cascade to chaos and a period-halving cascade to
the equilibrium point in the case of two time delays if the di¤erence between
two the delays is large.
Keywords: single and multiple …xed time delays, Hopf bifurcation, bounded
rationality, stability switch, nonlinear monopoly, gradient dynamics
JEL Classi…cation: C62, C63, D42

The authors highly appreciate …nancial supports form the Japan Society for the Promotionof Science (Grant-in-Aid for Scienti…c Research (C) 21530172), Chuo University (Joint Research
Grant 0981) and the Japan Economic Research Foundation. The research leading to this paper
started when the …rst author visited the Department of Systems and Industrial Engineering of the
University of Arizona and …nished when the second author visited the Department of Economics,
Chuo University. They appreciated hospitalities of those universities over their stay. The usual
disclaimer applies.
1

2

AKIO M ATSUM OTO AND FERENC SZIDAROVSZKY

1. Introduction
Traditional monopoly market has a structure in which there is only one seller,called a monopolistic …rm, and there are in…nitely many small buyers whose behavior is described by the market demand function. Further, the …rm is assumed
to be rational in a sense that full information or knowledge on demand is available
to it without any delays. As a result, it is enough for the …rm to select either priceor quantity (but not both) to maximize pro…t. Once one of price and quantity is
chosen, then the other is automatically determined through demand. No matter
which choice is made, it can jump to the same optimum point in one shot without any dynamic considerations. Full information is, thus, the most important tomake an optimal decision and to realize it. However, at the same time, it is verydi¢ cult, expensive and time-consuming to obtain. In reality, the monopolistic …rm
has only limited information. In addition to full information, most existing studies also assume the availability of instantaneous information. This may be due tomathematical convenience and is against actual observations that a seller usuallyobtains price information and sales data with some delays and reacts cautiously.
Getting closer to the real world and making the monopoly theory more convincing,
we replace this extreme assumption with the more plausible behavior assumption
in which full and instantaneous information is eliminated. First, it is assumed thatthe monopolistic …rm knows only a few points on the demand. A natural consequence of this alternation is that the …rm is not able to move to the optimal pointwith one shot but gropes for it. Second, it is assumed that in revising the decisions, the …rm experiences time delays inherent in phenomena like information andimplementation delays. Such a …rm is called boundedly rational. Therefore, we aim
at studying a dynamic model of a boundedly rational monopolistic …rm in order todescribe an adjusting process of optimal decisions with time delays.
In constructing dynamic economic models, the most common processes are basedon either the gradients of the pro…t functions or the best replies of the economic
agents. Time delays can be modeled in two di¤erent ways: …xed time delays andcontinuously distributed time delays. The choice of "time", continuous time or
discrete time, remains matter for debate. Cournot oligopoly is frequently discussed
in a discrete-time model with the best replies. In the case of the gradient method, we
mention the works of Bischi and Naimzada (1999) and Bischi and Lamantia (2002)
for discrete-time oligopoly dynamics without time delays. Recent developments on
the oligopoly theory can be found in Bischi et al. (2010) which contains models
with both discrete and continuous time scales. Concerning discrete-time monopoly
dynamics, we mention Puu (2003) and Naimzada (2011). The former assumes
bounded rationality and shows that the monopolistic …rm behaves in an erratic
way. The latter exhibits that a …xed delay dynamics of the monopoly with bounded
rationality can be described by the well-known logistic equation when the …rm takes
a learning activity of revising decisions.
In recent years it has been recognized in continuous-time economic dynamics
that a delay di¤erential equation is useful to describe the periodic and aperiodic
behavior of economic variables. Howroyd and Russel (1984) detect the stability
conditions of delay output adjustment processes in a general N -…rm oligopoly with…xed time delays. On the other hand, Chiarella and Khomin (1996) and Chiarella
and Szidarovszky (2001) examine delay di¤erential oligopolies with best replies
by using continuously distributed time delays. Furthermore, Bélair and Mackey

NONLINEAR DELAY M ONOPOLY W ITH BOUNDED RATIONALITY

3(1989) develop a model of price adjustment with production delays, Invernizzi andMedio (1991) investigate various economic models with continuously distributed
delays and con…rm analytically as well as numerically the conditions for chaotic
solution. Recently, Matsumoto (2009) reconstructs Goodwin’s accelerator model
as a delay neutral di¤erential equation and makes it clear that multiple limit cycles
coexist whereas Matsumoto and Szidarovszky (2011a) introduce a …xed delay inproduction and a mound-shaped production function into the neoclassical onesector growth model and show the birth of complex dynamics. With the in…nite
dimensionality created by a …xed-time delay, even a single …rst-order equation istransformed into an equation with a su¢ cient number of degrees of freedom topermit the occurrence of complex dynamics involving chaotic phenomena. This…nding indicates that …xed-time delay models of a dynamic economy may explain
various complex dynamic behavior of the economic variables. In addition to this,dynamics of a delay di¤erential monopoly with the gradient method has not yet been
revealed in the existing literature. Therefore, in this study, we draw our attention
to dynamics of a continuous-time and …xed-delay monopoly under a circumstance
where a …rm is boundedly rational and uses the gradient of the expected pro…t torevise its output decision.
The paper is organized as follows. Section 2 constructs a basic monopoly modelwith linear price and cost functions and then introduces time delay; single timedelay in the …rst part and multiple time delays in the second part. Local stability
conditions are derived and the birth of a Hopf bifurcation is studied. Section 3
performs numerical simulations to con…rm global behavior of a unstable monopoly
equilibrium. Section 4 concludes the paper.
2. Delay Monopoly
Consider optimal behavior of a boundedly rational monopolistic …rm which produces output q with marginal cost c. The price function is linear
f (q) = abq; a; b > 0:

We con…ne attention to a situation where the …rm can estimate the derivative of the
expected pro…t by using actual prices and demands it received in the past, although
it does not know the price function and does not even know that it is linear. The
estimated derivative at a value q e of output is assumed to be
d e
=a
dq e

2bq e :

c

So the approximating gradient dynamics is(2.1)

q_ = (q)d e
dq e

where (q) is an adjustment function and the dot over a variable means a time
derivative. In constructing best response dynamics, global information is required
about the pro…t function, however, in applying gradient dynamics, only local information is needed. We make the familiar assumption that the adjustment function
is linear in output:
Assumption 1.:(q) = q with

> 0:

4

AKIO M ATSUM OTO AND FERENC SZIDAROVSZKY

The gradient dynamics under Assumption 1 with an expected output is presented
by
(2.2)

q(t)
_ = q(t) [ac2bq e (t)]where t denotes a point of continuous time.
2.1. Single Fixed-Delay. Dynamics depends on the formation of expectations.
We start with a simple case in which at time t; the …rm forms its expected demand
to be equal to realized demand at time t; > 0: Since it can be supposed thatthe actual output produced at time t
is equal to relaized demand at the sametime, the expected demand is given in terms of output:
Assumption 2: q e (t) = q(t):
eReplacing q (t) in equation (2.2) with q(t), we obtain the output adjustment
process as a nonlinear di¤erential equation with one …xed-time delay:
(2.3)

q(t)
_ = q(t) [a

c

2bq(t

)] :

Equation (2.3) 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 a positive stationary output. We may call q M amonopoly equilibrium.
Our …rst concern is upon local stability of q M . For this purpose, we analyze
the linearized version of (2.3) about the monopoly equilibrium. Linearization andintroduction of the new variable x(t) = q(t) q M reduce equation (2.3) to
(2.4)

x(t)
_
=

x(t

) with= (ac) > 0;M

which has the zero-solution, x(t) = 0 (or q(t) = q ) for all t
0. If there is notime delay, = 0, then equation (2.4) becomes an ordinary di¤erential equationand has a stable solution, x0 e t ; where x0 is an initial point. If there is a positive
time delay, > 0, then equation (2.4) becomes a linear delay di¤erential equation
where initial data are given by a continuous function de…ned fort
0. It
is impossible to derive an explicit solution even though the equation seems to be
simple. In spite of this inconvenience, it is possible to analyze qualitative aspects ofthe solution in the following way. Substituting the exponential solution x(t) = x0 e t
into (2.4) yields the following characteristic equation associated with the linearized
delay equation:
(2.5)

+ e

= 0:

The su¢ cient condition for local asymptotical stability of the zero-solution is thatthe real parts of the eigenvalues are negative.
The characteristic equation is a function of and so are its roots. It is seenthat the zero solution is asymptotically stable for = 0. By continuity we expectthat it remains asymptotically stable with small values of : Equation (2.5) implies
that the real roots are at the intersection of an exponential curve (i.e., e) and anegative-sloping straight line (i.e.,= ). We have two real roots, one real root or
no real root, depending on the value of : It can be shown that the line is tangent
to the curve at= 1=e. For satisfying the tangency condition, equation (2.5)
generates a unique negative root. Further increases of ; ceteris paribus, may change
the zero real part of to positive making the system unstable. Such phenomena is

NONLINEAR DELAY M ONOPOLY W ITH BOUNDED RATIONALITY

5

referred to as a stability switch. In order to understand the stability switch of (2.4),
it is crucial to determine a threshold value =at which the real parts of thecomplex roots of (2.5) are zero and their derivatives are positive. We then assume,
without loss of generality, that = i ; > 0; is a root of (2.5) for = . In orderto hold equation (2.5) under > 0, the real and imaginary parts must satisfy thefollowings:
cos
=0
and
sin

= 0:
2

Adding up the squares of both equations yields

2=from which we obtain= :The necessary and su¢ cient conditions for the existence of a pair of purely imaginary roots are
cos
=0
and
sin

=

=1

from which we have in…nitely many solutions,
(2.6)

=

1
2

+ 2n

; n = 0; 1; 2; :::The real parts of the roots are zero for satisfying (2.6). We now detect thestability switch at which the equilibrium loses stability. Since is a function of delay; we need the minimum solution of for which a derivative of ( ) is positive.
By selecting as the bifurcation parameter and di¤erentiating the characteristic
equation (2.5) with respect to yield
(1

e

)

d
=
d

e

:

which is reduced to
d
d

1

=

1

e
e

:

Obtaining e== from (2.5) and substituting it into the last equation, we
can show that the real parts are positively sensitive to a change in ;
"
#
1
d
1
d(Re )
Re= 2 > 0 or
> 0:
d
d
=i
=i

This inequality implies that all roots that cross the imaginary axis at i cross fromleft to right as increases. Matsumoto and Szidarovszky (2011b) demonstrate thatstability switch does not occur for any > =(2 ) given in (2.6), although thereal parts are zero and their derivatives with respect to are non-zero. Hence thethreshold value of is=2This result is summarized as follows:

6

AKIO M ATSUM OTO AND FERENC SZIDAROVSZKY

Theorem 1. The delay output adjustment process (2.3) has a threshold valueof delay: The monopoly equilibrium is locally asymptotically stable for 0 < < ;locally unstable for >and undergoes a Hopf bifurcation at =whereisde…ned by
=

2 (a

c)

:

Theorem 1 is visualized in Figure 1(A) where the higher hyperbolic downward
sloping curve is the locus of= =2 and the lower hyperbolic curve is the locusof= 1=e: The latter curve is a partition curve between the real roots and thecomplex roots. The former curve is the stability switch curve that divides the …rst
quadrant of the the ( ; ) plane into two regions: the gray region in which themonopoly equilibrium is locally stable and the white region in which it is locallyunstable. Although local stability (instability) implies global stability (instability)
in a linear model, this is not necessarily a case in nonlinear dynamic system such as(2.3)1 . To observe global behavior of the unstable equilibrium, we perform numericalsimulations. Taking= 1=2; a = 3; c = 1; b = 1 and = 1; we select fourparametric combinations of ( ; ) indicated by the black points along the verticalline at= 1 in the white region of Figure 1(A). Corresponding dynamics aredescribed by limit cycles with di¤erent amplitudes as shown in Figure 1(B). It canbe seen that the amplitude becomes larger as the delay becomes longer.

(A) Divisions of the ( ; ) plane(B) Birth of limit cycles

Figure 1. Dynamics with one delay
2.2. Multiple Fixed-Delays. We next examine stability of the monopoly equilibrium when the estimate of the price function is based on a weighted average ofpast estimates at two di¤erent times, t
1 > 0 and t
2 > 0, which is the sameas estimating the derivative of the price function with a linear combination of two
1 Using a logistic single species population model, which is essentially the same as the delay
monopoly model (2.3), Wright (1955) shows global stability under the condition of < 3=2 andpresumes that his method can be used to extend the global result to < 2 = . However thisconjecture still remains open.

NONLINEAR DELAY M ONOPOLY W ITH BOUNDED RATIONALITY

7

past output values. Similarly to the case of a single …xed delay we have now thefollowing assumption:
Assumption 3: q e (t) = !q(t!)q(t1 ) + (12 ) with 0 < ! < 1:
Replacing q e (t) in equation (2.2) with this weighted average, we obtain the output adjustment process as a di¤erential equation with two …xed-time delays,
(2.7)

q(t)
_ = q(t) [ac2b (!q(t1 ) + (1!)q(t2 ))] :

M

Notice that q is also a positive stationary state of (2.7). As in the single delay case,
we …rst analyze local stability of q M : Linearizing equation (2.7) in a neighborhoodof the stationary point and using x(t) = q(t) q M yield a linear delay equation,
(2.8)

x(t)
_
=

[!x(t
t

Looking for a solution x(t) = x0 e
+ !e1 ) + (1

!)x(t

2 )]:

presents the characteristic equation
1

+ (1

!)e

= 0:

2

For notational simplicity, let
;

=

1 =

1 and

2 =

2

!)e

2

= 0:

which lead to the simpli…ed form,
+ !e

1

+ (1

Dropping the bar from ; we obtain the normalized characteristic equation,2
(2.9)

+ !e

1

+ (1

!)e

2

= 0:

When 1 = 2 , equation (2.9) is reduced to equation (2.5). Thus 1 6= 2 isassumed henceforth. Three cases are identi…ed: (1) ! > 1=2; (2) ! = 1=2 and (3)! < 1=2: Since the …rst and third cases are symmetric, we con…ne attention to thecase of ! 1=2.
We construct the loci of ( 1 ; 2 ) on which the real part of an eigenvalue is zero.
We thus assume that = i with > 0 is a root of equation (2.9) satisfying(2.10)

i + !e i

1

!)e i

+ (1

= 0:

2

Real and imaginary parts are(2.11)! cos(

1 ) + (1!) cos(2) = 0and(2.12)! sin(

1)(1!) sin(2 ) = 0:2 Suppose that the characteristic equation is

+ Ae

1

+ Be

2

=0

where the weights A and B are positive constants and A + B 6= 1: Dividing it by (A + B) yields(A + B)+Ae(A + B)

1+

B
e(A + B)

2

= 0:

Let
=

(A + B);i = (A + B)

i, ! =A;
A+B

then the resultant characteristic equation has essentially the same form as the one in (2.9).

8

AKIO M ATSUM OTO AND FERENC SZIDAROVSZKY

When the real part of an eigenvalue changes sign, then the corresponding point( 1 ; 2 ) has to be on the loci satisfying (2.11) and (2.12). Then we call it a stabilityswitch curve. As derived in Appendix I, it consists of the two segments, one segment
is denoted by L1 (k; n) and the other by L2 (k; n) for each k; n 2 N:
8
2
+ 2! 1
1
>
1
>
sin
+ 2k
=
>
1
>
<
2 !
(2.13)
L1 (k; n) :>2>
1
2! + 1
>
>
sin 1
+ 2n
: 2=
2 (1 !)

and(2.14)

L2 (k; n) :

8
>
>
>
>
<
>
>
>
>
:

1 =

2 =

1

sin 1

1

sin 1

2

+ 2!
2 !

1

+ 2k

2

2! + 1+ 2n
2 (1 !)

Specifying the value of !, we examine the shapes of L1 (k; n) and L2 (k; n):1Asymmetric weights: ! > :2
Using equations (2.13) and (2.14), we can compute the starting point ( s1 ;and the end point ( e1 ; e2 ) of L1 (k; n) :
s
1 =

e
1 =

1
2!
2

1

2

+ 2k andand the starting point ( S1 ;
S
1 =

E
1 =

1

e
2 =

and

2

s
2 =

1
2!

1

2

+ 2k

+ 2k and

E
2 =

and

2

3
+ 2n
2

;

+ 2n ;

SE2 ) and the end point ( 1 ;

1
2!
2

+ 2k

s
2)

S
2 =

E
2 ) of L2 (k; n) are

1
2!

1

2

+ 2n

;

+ 2n :

p
Notice that S2 is infeasible at= 2! 1; only thep n = 0: Since 2 = 0 for
segment of L2 (0; 0) between 2! 1 and 1 is feasible. Notice also that ( e1 ; e2 ) =
E
s
s
S
S
( E
1)) with …xed k: The …rst equality
1 ; 2 ) and ( 1 ; 2 ) = ( 1 ; 2 + 2 =(2!
implies that L1 (k; n) and L2 (k; n) have the same end point while the second equality
indicates that the starting point of L1 (k; n) is the same as that of L2 (k; n + 1).
Therefore the L1 (k; n) and L2 (k; n) segments form a continuous curve with n =0; 1; 2::: It can be con…rmed that both segments shift upward as n increases withgiven k and rightward as k increases with given n: We summarize these propertiesof L1 (k; n) and L2 (k; n) as follows:
Proposition 1. (1) Given k and n; the end point of L1 (k; n) is the same as theend point of L2 (k; n) while the starting point of L1 (k; n) is the same as the starting
point of L2 (k; n + 1): (2) The loci shift upward if n increases and k is …xed whereas
it shifts rightward if k increases and n is …xed.

NONLINEAR DELAY M ONOPOLY W ITH BOUNDED RATIONALITY

9

The monotonicity of the loci, L1 (k; n) and L2 (k; n); as well as their extreme(minimum and maximum) values can be examined based on the derivatives @ 1 =@and @ 2 =@ , which are presented in Appendix II. It is derived there that for j = 1; 2;#"@ 1= sign [ 1 + tan( 2 )]sign
@ Lj (k;n)
and

"

#
@ 2
sign
=
@ Lj (k;n)

sign [

2 + tan(

1 )]In order to simplify further discussions we change the coordinates ( 1 ;to get the transformed segments:
8
2+ 2! 1>1>+ 2k>
1 = sin><2 !
`1 (k; n) :>2>
2! + 1
>
>
+ 2n
sin 1
:
2 =
2 (1 !)
and`2 (k; n) :

8
>
>
>
>
<
>
>
>
>
:

sin 1

1 =

2

+ 2!
2 !

1

2 ) to (

1;

+ 2k

2

2! + 1+ 2n
2 (1 !)
They also form continuous curves with …xed k for n = 0; 1; 2; :::, which are periodic
in both directions 1 and 2 : Figure 2(A) shows these segments with k = 0 and
n = 0; 1:
In order to see the monotonicity of the loci we also show the curves
1 =2 = sintan(

12 ) and

2 =

tan(

1)

in Figure 2(A) where the former de…ned for the three domains of 2 ; ( =2; 3 =2);
(3 =2; 5 =2) and (5 =2; 7 =2) is illustrated by the ‡atter real curves and the latterde…ned for1 2 ( =2; ) is by the steeper curve. Let us see the monotonicityof L1 (0; 0) …rst. Locus `1 (0; 0) has an intercept denoted as m0 with the curvetan( 2 ) for1 =
2 2 ( =2; 3 =2). Before this intercept (in the directionof increasing value of which is indicated by the arrows) 1 decreases in , after
the intercept, increases. Therefore this point corresponds to the minimum valuetan( 1 )of 1 on L1 (k; n); denoted by m
2 >1 in Figure 2(B). On `1 (k; n),everywhere, so 2 decreases in here. Consequently, the slope of L1 (0; 0) de…nedby@ 2@ 2 =@=@ 1@ 1 =@is positive along the corresponding part from S0 to m0 of L1 (0; 0) and the inequality
is reversed along the remaining part from m0 to E0 as shown in Figure 2(B). We
now turn attention to the monotonicity of the L2 (0; 0) curve. Locus `2 (0; 0) has anintercept with the curve 1 = tan( 2 ) for 2 2 ( =2; =2): Since its ordinate
is negative, the intercept is not shown in Figure 2(A). It is checked that @ 1 =@ < 0and @ 2 =@ > 0; which then imply @ 2 =@ 1 <0 along the L2 (0; 0) locus.

2)

10

AKIO M ATSUM OTO AND FERENC SZIDAROVSZKY

The starting point S0 and end point E1 of the L2 (0; 1) locus are given by3;1) 2(2!

2(2!

1)

and

2

;

5
2

where the ordinates of these points are the same
3
2(2!

1)

=

5
if ! = 0:8:
2

Since the `2 (0; 1) locus intercepts once with thetan( 2 ) curve at point
1 =M and twice with the 2 = tan( 1 ) curve at points A and B, the corresponding
curve of L2 (0; 1) takes non-monotonic shape. Let see this more precisely. As
increases from 2! 1 to 1, the point ( 1 ; 2 ) moves along the `2 (0; 1) locus from
S0 to E1 . Before intercept M ,tan( 2 ) showing that 1 increases in ,1 <and after this intercept, 1 decreases, so this point corresponds to the maximum
value of 1 on L2 (k; n); denoted by Mtan( 1 ) in the
2 <1 : Since we haveleft of the 2 = tan( 1 ) curve and 2 > tan( 1 ) in the right, 2 decreases
between the two intercepts A and B, and increases otherwise. Therefore the …rst
intercept corresponds to the maximum value of 2 and the second corresponds tothe minimum value of 2 on L2 (k; n). By the three intersections between S0 andE1 , the whole L2 (0; 1) locus is divided into four pieces, along each of which we havethe following slopes in the ( 1 ; 2 ) plane,
@ 2
@ 1
@ 2
(2)
@ 1
@ 2
(3)
@ 1
@ 2
(4)
@ 1

(1)

@ 1
@ 2
> 0 and> 0 between S0 and M;
@
@
@ 1
@ 2
< 0 as< 0 and> 0 between M and A;
@
@
@ 1
@ 2
> 0 as< 0 and< 0 between A and B;
@
@
@ 1
@ 2
< 0 as< 0 and> 0 between B and E1 :@@> 0 asIn the same way, we can depict the loci of L1 (k; n) and L2 (k; n) for any othervalues of k and n. Notice that all the segments of L1 (k; n) and L2 (k; n) are locatedM
within the interval [ m1 ; 1 ]. In Figure 2(B) with the parameter values speci…ed
above,
m
1 ' 1:493 andM
1 ' 2:733:

In Figure 2(A), the maximum and minimum values ofm m
1 ' 1:427 and1 on `i (k; n) are given byM M
1 ' 1:715

NONLINEAR DELAY M ONOPOLY W ITH BOUNDED RATIONALITY(A) In the (1;2 ) plane(B) In the ( 1 ;112 ) plane

Figure 2. Partition curves
We now turn our attention to the stability of the monopoly equilibrium. Needlessto say, q M is asymptotically locally stable when 1 = 2 = 0: By continuity, it islocally asymptotically stable if 1 and 2 are su¢ ciently small. Since the segmentsMof L1 (k; n) and L2 (k; n) are located within the interval [ m1 ; 1 ], the vertical linemwith 1 < 1 crosses none of these segments. Therefore the output adjustment
process is asymptotically stable regardless of the values of 2 > 0 if 1 is strictly lessthan m
1 : Such delays of ( 1 ; 2 ) are called harmless and belong to the light-grayrectangle in the left of Figure 3.
Next we investigate stability for 1 > m
1 and the occurrence of Hopf bifurcation.
Fixing 2 > 0; we choose 1 as a bifurcation parameter and then di¤erentiate thecharacteristic equation (2.9) having = ( 1 ) with respect to 1 to obtain
d
+ !e
d 1

d
d 1

1

+ (1

1

!)e

2

d
d 1

2

= 0:

Solving this for the inverse of d =d 1 gives
d
d 1

1

=

1

! 1e

1

1

! 1e

1

!e

(1

!) 2 e211)+ 2 ( + !e
1
!e
1+ 2
!( 2
1)
+
=
1
!e
!
where we used the characteristic equation (2.9). If = i ; then

=

d
d 1

1

1+i 2
i !(cos( 1 ) i sin( 1 ))sin( 1 ) + 2 cos( 1 )=!
so Re(d =d 1 ) has the same sign as sin( 1 )+ 2 cos( 1 ): With any …xed positive
value of 2 we can gradually increase the value of 1 from zero until it intercepts
with a segment L1 (0; n) or L2 (0; n): It is easy to prove that at this and each laterintercepts there is a unique pure complex eigenvalue.
Re

=

Re

12

AKIO M ATSUM OTO AND FERENC SZIDAROVSZKY

Assume …rst that 1 crosses a segment L1 (0; n) from the left. There1 2(0; =2) implying that both sin( 1 ) and cos( 1 ) are positive as well as Re(d =d 1 ) >0. Therefore stability is lost everywhere on the segments L1 (0; n): Assume next that
0 as
1 crosses a segment L2 (0; n) from the left where cos(
1)
1 2 [ =2; ].
Stability is lost when Re(d =d 1 ) > 0 at the intercept and the real part of an
eigenvalue changes sign from positive to negative if Re(d =d 1 ) < 0: If there wasonly one stability switch from negative to positive real part before, then stability
is regained, since at the intercept with L2 (0; n) the real part of the same eigenvaluechanges back to negative. If there are more than one stability switches from negative to positive real parts before, then stability cannot be regained, since at least
one eigenvalue will still have positive real part. Based on (4.13), it is clear that
Re

d
d 1

=
=i

cos(
!

1)

@ 2
;
@ L2 (k;n)

therefore Re(d =d 1 ) > 0 when 2 increases in and Re(d =d 1 ) < 0 when 2
decreases in . Hence stability is switched to instability when @ 2 =@ > 0 whereas
instability might be switched to stability when @ 2 =@ < 0. Figure 3 illustrates
these segments of Li (k; n) for i = 1; 2 with the parameter values of = 0:5; a =3; b = c = 1, ! = 0:8; k = 0 and n = 0; 1; 2 when increases from 2! 1 to 1.3
The points of L1 (0; n) form the red curves, and those of L2 (0; n) the blue curves
while the green dotted points denoted as Si and Ei are the starting and end pointsof L1 (k; n) and L2 (k; n) as indicated by the directions of the arrows.
When the real part of an eigenvalue changes from negative to positive, thenstability is lost for sure. If the change is from positive to negative, then stability
may be regained but not necessarily, as it is illustrated in the following two cases.
(1)
If 2 = 2 as shown in Figure 3, then the …rst intercept is with L1 (0; 2) where
stability is lost, since the real part of an eigenvalue becomes positive. If 1 isincreased further, then there is a second intercept with L2 (0; 2) where the realpart of the same eigenvalue becomes negative again, so stability is regained. If
1 is increased even further, then there is a third intercept with L1 (0; 1) where(2)stability is lost again. However if we select 2 = 2 , then we also have threeintercepts, the …rst two are with L1 (0; 2) and L1 (0; 1); so two eigenvalues will havepositive real parts, and the sign of only one of them changes back to negative at thethird intercept with L2 (0; 2): Therefore in this case no stability switch occurs since
stability cannot be regained. The stability region is the union of the light-gray anddark-gray areas. Hopf bifurcation occurs on the boundary curve between the grayregion and the white region except where 2 is minimal or maximal on L2 (0; n);which are not really crossing points.
If the value of 1 is …xed and 2 is the bifurcation parameter, then by using thesame approach, the same stability region is derived and Hopf bifurcation occurs
Mat the crossing points except if 1 = m
1 =1 or
1 . These points are notcrossing points, since the corresponding vertical line is the tangent line to L1 (0; n) orL2 (0; n): This is the same result which was obtained earlier by Hale and Huang(1993) and Matsumoto and Szidarovszky (2011c) and summarized as follows:

3 We will soon refer to the two black dots denoted by A and B.

NONLINEAR DELAY M ONOPOLY W ITH BOUNDED RATIONALITY

13m
Theorem 2. (1) Delays are harmless if 1
0: (2) L1 (0; n) and
1 and 2L2 (0; n) constitute a stability-switch curve; The monopoly equilibrium is locally asymptotically stable in the dark-gray region of Figure 3 and it is locally unstable in
the white region.

Figure 3. Stability-switch curve in the( 1 ; 2 ) plane with k = 0
1
Symmetric Weights: ! = :2
Substituting ! = 1=2 into (2.13) and (2.14) we obtain the loci with symmetricweights,4
8
1>
sin 1 [ ] + 2k>
< 1=
(2.15)
L1 (k; n) :>>1:
sin 1 [ ] + 2n
=
2

and

(2.16)

L2 (k; n) :

8
>
>
<
>
>
:

1 =

2 =

1

1

sin 1 [ ] + 2k
sin 1 [ ] + 2n:

Clearly has to be in the unit interval in order to have feasible solution.
Loci L1 (k; n) and L2 (k; n) for k = 0; 1 and n = 0; 1 are depicted as the red
curves and the blue curves in Figure 4. When k = n = 0; these two loci constructa hyperbolic curve passing through the point ( =2; =2) which is the common end
point of L1 (0; 0) and L2 (0; 0). The curve divides the ( 1 ; 2 ) plane into two regions,a stable gray region in which all roots of the characteristic equation have strictly
negative real part for ( 1 ; 2 ) and a unstable white region in which the monopoly
equilibrium becomes locally unstable, which can be proved similarly to the nonsymmetric case. Notice that the curve is symmetric with respect to the diagonal
4 Hale (1979) determines the geometry of the stability region with ! = 1=2 in a di¤erent way.

14

AKIO M ATSUM OTO AND FERENC SZIDAROVSZKY

and asymptotic to the lines i = 1, since sin 1 ( )= converges to 1 as
! 0:This implies that the monopoly equilibrium is locally asymptotically stable for any
1 for i; j = 1; 2 and i 6= j; in other word, the delays are harmless.
i > 0 if j
We have the similar results as Proposition 2 in the case of ! = 1=2:
Proposition 2. (1) Given k and n; L1 (k; n) and L2 (k; n) de…ned on the unit
interval [0; 1] have the same end point. (2) Fixing k and increasing n shift the lociupward, while …xing n and increasing k shift the loci rightward. (3) Increasing k
and n together shifts the loci along the diagonal line, 1 = 2 :Stability results in the symmetric case are summarized as follows:
Theorem 3. (1) Delays are harmless if i > 0 and j
1 for i; j = 1; 2 andi 6= j: (2) L1 (0; 0) and L2 (0; 0) constitute a stability-switch curve, below which themonopoly equilibrium is locally asymptotically stable and above which it is locallyunstable.

Figure 4. Divisions of the ( 1 ;1when ! =22 ) plane

3. Numerical Simulations
In this section we perform simulations, …rst, to con…rm the analytical results
obtained above and, second, to see how the two delays a¤ect global dynamic behavior. Throughout this section we take = 0:5; a = 3; b = c = 1 under which= 1 always implying i = i for i = 1; 2:Harmless delays:m
We examine harmless delays of 1 and 2 such that i
0 and jj fori; j = 1; 2 and i 6= j: Two combinations of ( 1 ; 2 ) are selected from this region.
Figure 5(A) illustrates a time trajectory of q(t) for 1 = 1:2 and 2 = 3 which are
ordinates at point A in Figure 3 and so does Figure 5(B) for 1 = 1:2 and 2 = 15;

NONLINEAR DELAY M ONOPOLY W ITH BOUNDED RATIONALITY

15

which are ordinates of point B. Both trajectories take the same initial function,(t) = 0:1 for t 0, exhibit initial disturbances, which are sooner or later disappear
and …nally converge to the monopoly equilibrium. The initial disturbance with thelarger delay of 2 is larger and more aperiodic than the one with the smaller delayof 2 : The numerical simulations con…rm the analytical result that delay 2 has nome¤ect on asymptotic behavior of output if 1
1 .

(A)

1 = 1:2 and

2 =3

(B)

1 = 1:2 and

2 = 15Figure 5. Convergnet trajectories when the delays are harmless
Bifurcation diagram:To examine global behavior of the unstable equilibrium, we …x the value of 1 andincrease 2 from 0 to 7 =(2(2w 1)); the upper bound of 2 in Figure 3. For eachvalue of 2 ; the delay equation (2.7) is solved to get a time series of q(t); q(t1)
and q(t
)
for
1000
t
2000:
The
extremevalues(i.e.,
local
maximaandlocal
2
minima) of the time series q(t) are plotted against 2 . The resultant extremumagainst-delay plot is called a bifurcation diagram with respect to 2 in a continuous
time dynamical system.
Example 1: 1 = 2 and w = 0:8
In this example, 2 increases along the dotted straight line starting at 1 = 2 inFigure 3. The monopoly equilibrium is stable for smaller values of 2 and loses
stability when 2 crosses the locus of L2 (0; 0) from below. However, it regains
stability when 2 arrives at the locus of L1 (0; 0): When 2 is large enough to crossthe locus of L2 (0; 1) from below, the monopoly equilibrium becomes unstable. Thistransition of dynamics is illustrated from a di¤erent view point as a bifurcation
diagram shown in Figure 6(A). The …rst switching to instability from stability
takes place at the left most dotted point, 2 ' 0:646.5 It can be seen that the
5 This threshold value is calculated as follows. Using the …rst equation of (2.14) and solving

2=

1 =

1

sin

1

2 + 2!

2 !

1

+ 2k

for yields ' 0:891; and then substituting it to the second equation of (2.14) gives
Another threshold values are obtained in the same way.

2 ' 0:646:

16

AKIO M ATSUM OTO AND FERENC SZIDAROVSZKY

stable equilibrium point bifurcates to a limit cycle having one maximum and one
minimum and the amplitude of the cycle gets larger …rst and then smaller as 2
increases. After rapid shrinking of the amplitude, instability switches to stability
at the middle dotted point ( 2 ' 5:441) and thus the trajectory converges to themonopoly equilibrium. At the right most dotted point ( 2 ' 7:697) a bifurcation
to a limit cycle occurs and the expansion phase starts again. Since we set theupper bound of 2 to 14 in this example, the equilibrium does not regain stability
anymore. We will come back later to the bifurcation cascade de…ned for 2 > 14
in Example 5 below. As indicated by the shape of the partition curves in Figure 3,
the alternation between stability and instability could repeat more frequently for
another value of 1 ; especially, for 1 closer to m
1 . Two limit cycles corresponding
to two di¤erent values of 2 (the blue cycle for 2 = 4 and the red cycle 2 = 12)
are illustrated in Figure 6(B).

(A) Bifurcation diagram(B) Phase trajectories

Figure 6. Dynamics in Example 1
Example 2:

1 = 2:3 and w = 0:8

In this example, 1 is taken to be 2:3 and 2 is increased from zero up to
7 =2(2w 1)(' 18:3). As seen in Figure 7(A), limit cycle is born when 2 enters into
the instability region and its amplitude becomes gradually larger as 2 increases.
The cycle rapidly becomes smaller when 2 approaches the stability-switching point.
The equilibrium is stabilized suddenly at the left most dotted point ( 2 ' 6:638)
and destabilized again at the right most dotted point ( 2 ' 8:041): The bifurcation
diagram in the second instability phase implies that the number of trajectory’s
extrema increases from two to four and then to eight. Further increases of 2
then decrease this number from eight to four and …nally to two. As a result of
increasing 1 in this example to 2:3 from the value 2:0 of the previous example,
the bifurcation process becomes a little bit complicated and the delay equation
generates a multi-period cycle. In Figure 7(B) a four-period cycle is illustrated for
2 = 11:NONLINEAR DELAY M ONOPOLY W ITH BOUNDED RATIONALITY(A) Bifurcation diagram

17(B) Phase trajectory

Figure 7. Dynamics in Example 2
Example 3: 1 = 2:5 and w = 0:8
We increase the value of 1 to 2:5 and take the upper bound of 2 as 7 =(2(2w1)) again. It can be seen in Figure 8(A) that the interval of 2 can be divided into
two instability phases. In the …rst phase in which 2 is less than 7:397, dynamic
behavior is almost the same as in the previous examples, although the monopoly
equilibrium is now already unstable for 2 = 0. Interesting dynamics appears in thesecond instability phase in which the monopoly equilibrium undergoes a numberdoubling cascade, a chaotic cascade and a number-halving cascade.6 The birth ofcomplicated dynamics could be due to a larger value of 1 . Figure 8(B) displays achaotic process of q(t1 ) and q(t) for 2 = 14:(A) Bifurcation diagram(B) Phase trajectory

Figure 8. Dynamics in Example 3
6 "number" means the number of the extrema.

18

AKIO M ATSUM OTO AND FERENC SZIDAROVSZKY

Example 4: 1 = 2 ' 1:57 and w = 0:5
In this example we change the value of w to 0:5 and examine dynamics in thecase of symmetric weights. Taking 1 = =2; we increase the value of 2 from0 to 9 =2 along the dotted vertical line passing through the points, ( =2; 5 =2)
and ( =2; 9 =2) in Figure 4. The monopoly equilibrium is locally asymptoticallystable for 2=2. After it loses stability, there is a number-increasing cascade.
In Figures 9(B) and (C), we display a one-period cycle with two maxima and two
minima and a two-period cycle with three maxima and three minima. As far as
the numerical simulations are concerned, only periodic cycles can be born in thesymmetric case.

(A) Bifurcation diagram(B) Phase trajectory(C) Phase trajectory

Example 5: 1 = 2; ! = 0:8 and 2 > 14
In the last example, we re-examine dynamics of Example 1 for 2 > 14: Figures
10(A) and (C) are continuations of Figure 5(A) for 2 > 14 and Figure 10(B)combines Figure 10 (A) and Figure 10(C). The only di¤erence between Figures (A)and (C) is the selection of the initial function for t < 0, q1 (t) = 0:1 in the former
and q1 (t) = 0:8 in the latter. The bifurcation diagrams are identical for 2 < 14:5;
apparently di¤er for 15 < 2 < 16 and then become identical again for 2 larger
than 16. Figure 10 implies the coexistence of attractors.

(A) (t) = 0:1(B) Combining
Figure 10. Dynamics in Example 5(C) (t) = 0:8

NONLINEAR DELAY M ONOPOLY W ITH BOUNDED RATIONALITY

19

4. Concluding Remarks
This paper constructed a delay monopoly model in which a monopolistic …rm
was boundedly rational and examined its stability/instability when dynamics was
driven by the gradient process. The paper was divided into three parts. In the…rst part, we assumed that the monopolistic …rm had one time delay in its production and detected the threshold value of the production delay at which stability
switched to instability. In the second part that was the main part of this paper, we
introduced two …xed-time delays into the formation of the expected demand. We…rst determined the shape of the stability-switch curve that divides the parametricspace of the delays into stable and unstable regions. We then showed that the monopoly equilibrium undergoes a Hopf bifurcation when the parameter combination
crosses the stability-switch curve. In the …nal part, we speci…ed parameter values
and numerically investigated …ve special cases in detail in which the global behaviorof the equilibrium was illustrated. In particular, it was demonstrated that a onedimensional di¤erential equation with two …xed-time delays could generate a wide
variety of dynamics ranging from a limit cycle to chaos with unstable monopoly
equilibrium.

Appendix I
In this Appendix I, we derive relations (2.13) and (2.14) from (2.11) and (2.12).
Moving the second term of (2.11) to the right hand side, squaring both sides andusing the well-known relation cos2 = 1 sin2 yieldsin2 (!2 1
Let x = sin(1 ) and y = sin(
2

! (1

1)

!)2 1

= (1

sin2 (

2):

2 ). Then the last equation is written as222x ) = (1

!) (1

y )

!)2 y 2 = 1

2!:which is arranged to be! 2 x2 + (1

(4.1)
Substituting x = sin(y yield1 ) and y = sin(2 ) into equation (2.12) and solving it for

!x
:
1 !
Substituting further (4.2) into (4.1) and solving the resultant equation for x give
(4.2)

y=

2

(4.3)

x=

+ 2!
2 !

1

which is substituted into (4.2) to obtain
2

(4.4)

2! + 1:2 (1 !)

y=Replacing x of (4.3) with sin( 1 ) and y of (4.4) with sin(meterized curve in the ( 1 ; 2 ) plane:
2

(4.5)

sin(

1) =

+ 2!
2 !

1

2

and sin(2) =2 ); we obtain a para-

2! + 1:2 (1 !)

20

AKIO M ATSUM OTO AND FERENC SZIDAROVSZKY

In order to have real roots, we need to restrict the range of each of the right handsides as2
2
+ 2! 1
2! + 1
1
1 and
1
1:
2 !
2 (1 !)
With > 0 and 1 > ! 1=2, these inequality conditions are reduced to2!
2

Since + 2! 1 > 0 for
in…nitely many solutions of
(4.6)

1

1 =

1

2 [2!
1:
2

sin 1

1:

1; 1]; solving the …rst equation in (4.5) yields
+ 2!
2 !

1

+ 2k

with k 2 N

or2+ 2! 1+ 2`
with ` 2 N
2 !
where N is a set of nonnegative integers.
We next determine 2 : Since 2 2! +p1 can have bothpsigns, we solve thesecond equation of (4.5) in two cases:
2! 1 and < 2! 1. In the …rst
case, 2 2! + 1 is nonnegative and thus sin( 2 )0 while in the second case2
2! + 1 < 0 and so sin( 2 ) < 0: Therefore in both cases we have in…nitely
many solutions of 2 in the form
1

(4.7)

1 =

(4.8)

2 =

1

sin 1

2

2! + 1+ 2m
2 (1 !)

sin 1

with m 2 Nor2

2! + 1+ 2nwith n 2 N:
2 (1 !)
Two solutions of 1 and two solutions of 2 produce four combinations. However,
only two combinations of ( 1 ; 2 ) are possible since the sign of cos( 2 ) must bedi¤erent from the sign of cos( 1 ) due to (2.11):

(4.9)

2 =

if

1

sin 1

1 is given by (4.6), then2 is given by (4.9)

andif 1 is given by (4.7), then 2 is given by (4.8).
Therefore we have two loci of ( 1 ; 2 ) for each k; n 2 N given in (2.13) and (2.14).

Appendix II
In this Appendix II, we will analyze the dependence of 1 and 2 on in order to examine the shape of the loci. Consider …rst a segment L1 (k; n). Simpledi¤erentiation shows that
2
2
@ 1
+2! 1
1
2!+1
= 12 sin 1+ 2k + 1 r
2 !
2 2!
2 +2! 1 2
@ L1
1
2 !
=

1
2

1+

1
2 cos(

1)

sin(

2)

1

!
!

NONLINEAR DELAY M ONOPOLY W ITH BOUNDED RATIONALITY

21

We use the …rst equation of (2.13) and (2.11) to obtain
@ 1
=
@ L1

(4.10)
Since cos(

2(

1 + tan(2 ))1 ) < 0 on L2 (k; n); we also have

@ 1
=
@ L2

(4.11)
Similarly
@ 2
@ L1

1

=

=

1
2

1
2

2

(

2
2!+1
2 (1 !)

sin 1

2+

1

1

sin(

2 cos(

2)

@ 2
=
@ L1

1

@ 2
=
@ L2

1

1 + tan(

+ 1r

+ 2n

1)

2 )) :

1
1

!
1

!

Therefore we have
(4.12)

2(

2 + tan(

1 ))

2

(

2 + tan(1 )) ;

and(4.13)
since cos(2 ) > 0 on L2 (k; n):

2 2!+12 (1 !)

2

2+2! 1
2 (1 !)

22

AKIO M ATSUM OTO AND FERENC SZIDAROVSZKY

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Current address : Akio Matsumoto: Department of Economics, Chuo University, 742-1, HigashiNakano, Hachioji, Tokyo, 192-0393, Japan.
E-mail address : akiom@tamacc.chuo-u.ac.jp
Current address : Ferenc Szidarovzky: Department of Systems and Industrial Engineering,
University of Arizona, Tucson, Arizona, 85721-0020, USA.
E-mail address : szidar@sie.arizona.edu