Dynamics of Boundedly Rational Monopoly
with Time Delay
Ferenc Szidarovszkyz
University of Arizona

Akio Matsumotoy
Chuo University

Abstract
The purpose of this paper is to study dynamics of a monopolistic …rmin a continuous-time framework. The …rm is assumed to be boundedly rational implying that it can obtain only limited information and experience
delay information on demand. A dynamic adjustment process is based on
the gradient of the expected pro…t. It is analytically demonstrated that amonopoly equilibrium undergoes a Hopf bifurcation when it loses stability.
Global dynamic behavior is con…rmed by numerical simulations.

1

Introduction

In the recent literature, it has been recognized in continuous-time economic dynamics that a delay di¤erential system is useful to describe the periodic andaperiodic behavior of economic variables. Time delays can be modeled in twodi¤erent ways: …xed-time delays and continuously distributed time delays. Invernizzi and Medio (1991) investigate various economic models with the latter andcon…rm analytically as well as numerically the conditions for chaotic solution.
On the other hand Matsumoto (2009) reconstructs Goodwin’s accelerator modelwith the former as a delay neutral di¤erential equation and examine the condition under which multiple limit cycles coexist. More recently, Matsumoto andSzidarovszky (2011a) introduce a production delay of both types and a moundshaped production function into the neoclassical one-sector growth model andshow the birth of complex dynamics. With the in…nite dimensionality created
The authors highly appreciate …nanical supports from Chuo University (Joint Research
Grant 0981), the Japan Society for the Promotion of Science (Grant-in-Aid for Scienti…c Research (C) 21530172) 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.
y Department of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo, 1920393, Japan. akiom@tamacc.chuo-u.ac.jp
z Department of Systems and Industrial Engineering, University of Arizona, Tucson, Arizona, 85721-0020, USA. szidar@sie.arizona.edu

1

by a …xed-time delay, even a single …rst-order equation is transformed into an
equation with a su¢ cient number of degrees of freedom to permit the occurrenceof complex dynamics involving chaotic phenomena. This …nding indicates that…xed delay models of a dynamic economy may explain various complex dynamic
behavior of the economic variables.
This study is an application of Matsumoto and Szidarovszky (2011b) wherea complete eigenvalue analysis is given for a certain class of di¤erential equations
with a single delay. In particular, it considers monopoly dynamics with a …xed
time delay. It is well known that traditional monopoly market has such a structure in which there is only one seller, called a monopolistic …rm or a monopoly,
and there are in…nitely many small buyers whose behavior is described by themarket demand function. Further, the …rm is assumed to be rational in a sensethat full information or knowledge on demand is available free of charge without
any delays. As a result, it is enough for the …rm to determine either price orquantity (but not both) so as to maximize pro…t. Once one of price or quantity is chosen, then the other is automatically determined through the demand
function. No matter which choice is made, it can jump to the same optimum
point in one shot without any adjustment considerations. In reality, however,
the monopolistic …rm has only limited information, usually obtains 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 whichfull and instantaneous information is eliminated. Indeed it is assumed …rst thatthe monopolistic …rm knows only a few points on the demand function (i.e.,
partial information). Second it is assumed that the …rm experiences time delays
inherent in phenomena like information and implementation delays (i.e., delay
information). Such a …rm is called boundedly rational. A natural consequenceof alternating behavioral assumptions is that the …rm is not able to arrive at the
optimal point with one shot but gropes for it. In addition to this, dynamics of adelay di¤erential monopoly with the gradient method has not yet been revealedin 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…tto revise its output decision.1
The paper is organized as follows. Section 2 is divided into two parts. In the…rst part, a gradient dynamic monopoly model with time delay is constructed
and then local stability is considered. It the second part, numerical simulations
are given to con…rm global stability of a unstable monopoly equilibrium. Section
3 concludes the paper.
1 Concerning

discrete-time monopoly dynamics, we mention Puu (2003) and Naimzada(2011). The former assumes bounded rationality and shows that the monopolistic …rm behavesin an erratic way. The latter exhibits that …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.

22

Delay Monopoly

Consider optimal behavior of a boundedly rational monopolistic …rm which
produces output denoted by q with marginal cost c. The price function isassumed to be linear
f (q) = a bq; a; b > 0:
We con…ne attention to a situation where the …rm does not know the pricefunction and does not even know that it is linear. It is however able to estimate
its derivative using actual prices it received in the past. By using numericaldi¤erentiation, it is able to estimate the derivative at a value q e of output which
is believed to be close to the actual output it would select. The estimatedderivative value isd e= a c 2bq e :dq e
So the approximating gradient dynamics is
q_ = (q)

d e
dq e

(1)

where (q) is an adjustment function. 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 has linear dependency on output:
Assumption 1.

(q) = q with> 0:

The gradient dynamics under Assumption 1 with an expected output ispresented byq(t)_ = q(t) [a c 2bq e (t)](2)
where t denotes a point of continuous time.

2.1

Single Fixed-Delay

Dynamics depends on the formation of expectations. We consider a simple casein 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 that the actual output
produced at time t
is equal to demand, the expected demand is given interms of output:
Assumption 2 q e (t) = q(t):

Replacing q e (t) in equation (2) with q(t ), we obtain the output adjustment
process as a nonlinear di¤erential equation with one …xed-time delayq(t)_ = q(t) [a

c

3

2bq(t

)] :

(3)

Equation (3) has two stationary points; a trivial point q(t) = 0 and a non-tirival
point
a c
qM =
2b
where a > c is assumed to ensure a positive stationary output. We may call q M
a monopoly equilibrium.
Our …rst concern is upon local stability of q M . For this purpose, we analyze
the linearized version of (3) about the monopoly equilibrium. Linearization andintroduction of the new variable q (t) = q(t) q M reduce equation (3) toq_ (t) =q (t) with= (ac) > 0;(4)which has the zero-solution, q (t) = 0 (or q(t) = q M ) for all t
0. If there
is no time delay,= 0, then equation (4) becomes an ordinary di¤erential
equation and has a stable solution, q0 e t ; where q0 is an initial point. If
there is a positive time delay, > 0, then equation (4) becomes a linear delaydi¤erential equation where initial data are given by a continuous function de…nedfor
t 0. It is impossible to derive an explicit solution even though theequation seems to be simple. In spite of this inconvenience, it is possible toanalyze qualitative aspects of the solution in the following way. Substituting
the exponential solution q (t) = q0 e t into (4) yields the characteristic equation
associated with the linearized delay equation
+ e

= 0:

(5)

The su¢ cient condition for local asymptotical stability of the zero-solution isthat the real parts of the eigenvalues are negative.
The characteristic equation is a function of and so are its roots. Since
the monopoly equilibrium is asymptotically stable for= 0; we expect, bycontinuity, that the equilibrium remains asymptotically stable with positive but
small values of : To examine the location of the roots, we multiply both sidesof equation (5) by and introduce the new variables, x =and A = : Thenequation (5) is simpli…ed to be
x + Ae x = 0:

(6)

Real roots of equation (6) can be checked by a graphical method using the twofunctions,xand y2 = e x :y1 =AReal roots correspond to intersections of these functions. Depending on thevalue of A; the two functions have two, one and zero intersections implying that
equation (6) has two, one or no real roots. Consider the condition needed forthe straight line to be tangent to the exponential curve. As is seen in Figure
1, let us denote the tangency point by P = (x; y) at which the following two

4

conditions are satis…ed:
(i)

1
=
A

(ii)

x
= e x : the ordinates of the two functions for x = x are equal.
Ae x : the slopes of the two functions are equal;

From (i) and (ii), x = 1. Therefore A = 1=e is the tangency condition andequation (6) has two distinct real roots if A < 1=e and a pair of conjugate
complex roots if A > 1=e:Figure 1. Tangency condition
Assume that x = + i withand arranging the terms give
+ Ae

cos

> 0:2 Substituting this root in equation (6)
+ if

Ae

sin g = 0

where the real and imaginary parts should be equal to zero,
+ Ae

cos

=0

(7)

Ae

sin

= 0:

(8)

and
If sin = 0; then from equation (8), = 0; which implies that the root is realand violates the assumption > 0. So we eliminate the case of sin = 0 or
2 If

+ i withroots with > 0:

> 0 is a root, then so also is the one with5

< 0: We consider only the

= n for n 2 N; the set of integers including zero. On the other hand if
sin 6= 0, then from (8),
e

=

A sin

:

Substituting it into equation (7) presents
=

cot

(9)

which is then substituted into equation (8) to obtain= Ae cot sin(10)where the right-hand side is denoted by f ( ) in the following. The imaginary
part of the eigenvalue is a solution of equation (10) and the real part is given
by equation (9).3 We de…ne two intervals of ,1 = [n2N

n ;

2

+n

and

2 = [n2N2+ n ; (1 + n)

Equation (9) implies that sign( ) = sign(cot ) where cot > 0 for 2 1
and cot < 0 for
2 2 . Thus the real parts are negative for2 1 andpositive for2 2 . Accordingly we call 1 the stable interval and 2 theunstable interval.
We are concerned with the location of solving equation (10). The graphsof f ( ) with A = =2 and A = 5 =2 are shown in Figures 2A and 2B where
they take a mound-shape pro…le for
2 (0; ), furthermore f 0 ( ) > 0 for0
2 ((2n 1) ; 2n ) and f ( ) < 0 for 2 (2n ; (2n + 1) ) for n 1.4 The
value of is an intersection of f ( ) with the 45 line. Since equation (6) istranscendental, it has in…nitely many solutions. In consequence, the number ofintersections is also in…nite. With the assumption > 0; the roots are purely
imaginary when cot = 0 or = =2 + n . However, to hold > 0; sin > 0
is needed from equation (10), so = 0 and > 0 only for=2+ 2n :

Substituting = 0 and this threshold value of into equation (8) yields = A;
which can be written as
A = + 2n :2
The determination of under A = =2 (i.e., n = 0) is illustrated in Figure 2A
where three red dots denote the intersections, e1 ; e2 and e3 . The values of
3 There are several ways to solve these two equations forand : We outline the two ways,
one provided by Frisch and Holme (1935) and the other by Hayes (1950) in the Appendix.
4 Needless to say, the shape of the graph can be analytically determined. See Matsumoto
and Szidarovszky (2011b).

6

and

are
0
1 =

2

at e1 and5
0
2 ' 7:65 <20
1 = 0;

at e2 and

0
2 '

1:60;

(11)

9
0
3 ' 13:98 <at e3 and 03 ' 2:20
2where the upper script indicates the value of n and the subscript corresponds
to the numbering of the intersection. Let xnj = nj + i nj ; then x01 is a purely
imaginary root and the real parts of x02 and x03 are negative implying that 02
and 03 are in 1 : Graphically, the intercepts e2 and e3 are in the shaded regionsof Figure 2(A).

Figure2(A) Determination of

for A = =2

If A < =2; then the 45 line crosses the = =2 vertical line above A, so<=2, 2 < 5 =2 and 3 < 9 =2 while the corresponding s are negative.
1
If A > =2; then the 45 line crosses the= =2 vertical line below A, so
>
=2.
The
valuesofanddependon the speci…c value of A and12
3
the directions of the inequality are undetermined. To see this dependency let
A = =2 + 2 (i.e., n = 1). In the same way as for A = =2; we can obtain theordinates of the three intersections denoted by the red dotted points in Figure

7

2B,
1
1 ' 2:09 >

5
1
2 =

2

2

at e1 and1
1 ' 1:19;

1
2 = 0;

at e2 and

(12)

9
1
3 ' 14:10 <at e3 and 13 ' 0:59:
2
Notice that x11 has positive real part, x12 is a purely imaginary root and x13 has
negative real part implying that 11 is in 2 and 13 is in 1 : Although it is noteasy to see the location of 13 in Figure 2B, it is in the shaded region.

Figure 2(B). Determination of

for A = 5 =2

It is also clear in general that that if A = =2 + 2n ; then the root xnn+1 haszero real part, nn+1 = 0 with imaginary part nn+1 = =2 + 2n . Furthermore
its real part becomes positive if A > =2 + 2n and negative if A < =2 + 2n .
We summarize these results:
Lemma 1 Delay di¤ erential equation (6) has a root with < 0 and 2 1 if
< =2 + 2n , a root with > 0 and 2 2 if
> =2 + 2n and a rootwith = 0 and = =2 + 2n if= =2 + 2n :Returning to the de…nition of A; the tangency condition is written as=
1=e on which there is a unique negative root. This is a partition curve between
the real and complex roots. With a given value of ; a small increase of makes> 1=e implying that equation (5) generates a pair of complex roots with8

negative real parts. Further increases of may change the negative real part ofto positive making the system unstable. Such phenomena is referred to as astability switch. In order to understand the stability switch of (5), it is crucial todetermine a threshold value =at which the real parts of the complex roots
are zero and their derivatives are positive. We then assume that = i ; > 0;
is a root of (5) for = . By Lemma 1, we have in…nitely many solutions withthe zero real parts when
=

1
2

+ 2n

; n2N

(13)

We now detect the stability 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 anddi¤erentiating the characteristic equation (5) with respect to yield(1ewhich is reduced to

)
1

d
d

=

d
=
d
1

e

e
e

:

:

Obtaining e== from (5) and substituting it into the last equation, we
can show that the real parts are positively sensitive to a change in ;"#1dd(Re )
1Re== 2 > 0:dd=i=iThis inequality implies that the real part turns to be positive from negative
when crosses the imaginary axis.
Lemma 2 All roots of equation (5) that cross the imaginary axis at i crossfrom left to right as increases,d(Re )d> 0:=i

Lemmas 1 and 2 imply that the real parts of all roots are negative for <=2 and the real part of one root becomes positive when > =2 . Therefore
stability switch occurs when crosses the = =2 curve. On the other hand,
it does not occur when crosses the = ( =2 + 2n )= curve with n > 1.
Although the real part of the root with = =2 + 2n is zero and its derivativewith respect to is positive, there are already roots with positive real parts.
Hence the threshold value of is=This result is summarized as follow:
92

Theorem 1 The delay output adjustment process (3) has a threshold valueofdelay: The monopoly equilibrium is locally asymptotically stable for 0 < < ;locally unstable for >and undergoes a Hopf bifurcation at =whereis the threshold value of de…ned by
=

2.2

2 (a

c)

:

Numerical Simulations

Theorem 1 is visualized in Figure 3 where the higher hyperbolic downward
sloping real curve is the locus of= =2 and the lower hyperbolic real curve isthe locus of= 1=e: The former curve divides the …rst quadrant of the ( ; )plane into two regions: the gray region in which the monopoly equilibrium islocally stable and the white region in which it is locally unstable. Although local
stability (instability) implies global stability (instability) in a linear model, thisis not necessarily a case in nonlinear dynamic system such as (3)5 . Theorem 1
and the following simulations analytically and numerically con…rm the birth ofa periodic cycle when a parametric combination of ( ; ) is selected from thewhite region of Figure 3. To examine qualitative properties of such a periodic
solution, we perform three numerical simulations in which a = 3; c = 1 andb = 1 are …xed whereas and/or are varied.

Figure 3. Division of the ( ; ) plane into
stable and unstable regions
5 Using a logistic single species population model, which is essentially the same as the delay
monopoly model (3), Wright (1955) shows global stability under the condition of < 3=2
and presumes that his method can be used to extend the global result to < 2 = . However
this conjecture still remains open.

10

In the …rst simulation, we change both values of and by keeping= 2.
In particular, four di¤erent points, (2=3; 3); (1; 2); (2; 1) and (3; 2=3) are selectedfrom the= 2 locus. These points are in the white (unstable) region andmarked by the circles in Figure 3. Numerical results with the initial function(t) = q M 0:1 for< t 0 are summarized as follows. Figure 4(A) depicts
the cyclic motions of delay equation (2) in the (q(t); q(t)) plane. Note that
although the four periodic motions corresponding to the di¤erent pairs of andare depicted there, only one spiral motion toward a limit cycle is observed. Thisimplies that no matter which pair of and is selected, the phase trajectories
have exactly the same periodic motion in the (q(t); q(t)) plane as far as
the product of and is the same. Time trajectories of q(t) from t = 0 to
t = 12 are illustrated in Figure 4(B).6 It is seen that the time trajectories havedi¤erent lengths of the period but …nally converge to the same periodic cycle.
Observing these numerical results, we …nd the following:
Numerical Result 1. with …xed value of,

1. the amplitudes of the limit cycles with di¤erent values of
same;

and

are the2. the time to arrive at the limit cycle becomes longer as

becomes longer;

3. the length of a period of the cycle becomes longer as

becomes longer.

(A) Phase trajectories(B) Time trajectories

Figure 4. Numerical result in the …rst simulation
In the second simulation, we take four di¤erent values of , points (1; 7=4); (1; 2);(1; 5=2) and (1; 3) from the dotted vertical line at = 1; in order to detect the
6 To prevent Figure 4(B) from getting clumsy, we limit the length of time period short.
This restriction is inessential.

11e¤ect on dynamic behavior caused by changing the length of delay . Thesefour points are marked by crosses in Figure 3. Figures 5(A) and 5(B) illustrate
four di¤erent limit cycles in the (q(t); q(t)) plane and four di¤erent time
trajectories of q(t) against time between t = 0 and t = 30. Note that samecolor of the trajectories means same parameter speci…cations. The result aresummarized as follows:
Numerical Result 2. As

becomes larger with …xed value of ,1. the amplitude of the cycle becomes larger;
2. the arrival time to the limit cycle becomes shorter;
3. the length of a period becomes longer.

(A) Phase trajectories(B) Time trajectories

Figure 5. Numerical results in the second simulation
In the third simulation, contrary to the second one, we take four di¤erent
values of , (7=4; 1); (2; 1); (5=2; 1) and (3; 1) from the dotted horizontal linewith = 1; in order to detect the e¤ect on dynamic behavior caused by changing
the value of . These four points are marked by triangles in Figure 3. Figures
6(A) and 6(B) illustrate four limit cycles in the (q(t); q(t)) plane and fourtime trajectories of q(t) against time between t = 0 and t = 30. Note thatsame color of the trajectories means same parameter speci…cations. We havethe following result:
Numerical Result 3. As

becomes larger with …xed value of ,1. the amplitude of the cycle becomes larger;
2. the arrival time to the limit cycle becomes shorter;12

3. the length of a period becomes longer.

(A) Phase trajectories(B) Time trajectories

Figure 6. Numerical results in the third simulation
Theorem 1 shows that the delay equation (2) gives rise to a limit cycle
when the monopoly equilibrium becomes locally unstable. Through numericalsimulations, we arrive at the following results concerning the sensitivity of thecycles to changes in the parameter speci…cations:
Summary of Numerical Simulations
1 the amplitude of the cycle becomes larger as

becomes larger;

2 the period of the cycle becomes longer as one of two parameters,
becomes larger and the other is kept to be constant;and ;3 the convergence time to the limit cycle becomes shorter as one of two parameters, and ; becomes larger and the other is kept to be constant.

3

Conclusion

We constructed a dynamic model driven by the gradient process in which amonopoly was boundedly rational and single delay was presented in the formation of the expected demand. Main analytical results were summarized in
Theorem 1 in which it was demonstrated that a monopoly equilibrium was
locally asymptotically stable when the delay is less than the threshold value; lost its stability when =and underwent a Hopf bifurcation. We also
simulated the model and numerically con…rmed the following:(N1) The monopoly equilibrium bifurcates to a limit cycle.
13

(N2) As the delay becomes larger, both the amplitude and the period of thecycle become larger.
Appendix
In this Appendix, we brie‡y summarize two di¤erent ways to solve equations(7) and (8) for and proposed by Frisch and Holme (1935) and Hayes (1950).
Although they consider the more general characteristic equation having the formx + B + Ae x = 0; we apply their approaches to our simpli…ed characteristic
equation, x + Ae x = 0.
Frisch and Holme (1935) start with equation (8) to obtain,
e =A

sin

which can be rewritten, by using logarithms, as= log A + log

sin

:

(A-1)

Substituting these relations into equation (7) yields
cot

+ log

sin

= log A 1 :Write the left-hand side by fF H ( ) so that, given A;(A-2)
is a solution offF H ( ) = log A 1 :The graphs of fF H ( ) with A = =2 and A = 5 =2 are illustrated in Figures
I(A) and I(B), respectively. It appears that each graph has three intersections
with the log A 1 line. The red points denoted as ei correspond to the coordinatesof the intersections. Their abscissas are the values of and values of are given
by equation (A-1). It can be con…rmed that the values of and are the sameas those obtained in (11) and (12).

14(A) A = =2

(B) A = 5 =2

Figure I. Frisch-Holme approach
Hayes (1950) rewrites equations (7) and (8) as
=

Ae

cos

= Ae

sin :

and
Dividing the …rst equation by the second and solving it forof the root as a function of=cot

give the real part(A-3)

which is the same as equation (9). Adding up the squares of both equations andsolving it for gives the imaginary part of the root as a function ofp2:= A2 e 2(A-4)

The right hand sides of (A-3) and (A-4) will will be denoted as fH ( ) and gH ( ).
The = fH ( ) curve is positive-sloping and the = gH ( ) curve is negative
sloping in Figures II(A) and II(B). The -values of the intersections of these
two functions are the solutions of = gH (fH ( )) and the -values are given
by = fH ( ): The good point of his approach is that the intersections are thecoordinates in the complex plane. So it is easy to see whether the real part isnegative or positive. The values of and at the intersections are the same as
those given in (11) and (12).

15(A) A = =2

(B) A = 5 =2

Figure II. Hayes approach16

References[1] Frisch, R., and H. Holme, "The Characteristic Solutions of a Mixed Di¤erence and Di¤erential Equation Occuring in Economic Dynamics," Econometrica, 3 (1935), 225-239.
[2] Hayes, N. D., "Roots pf the Transcendental Equation Associated with a
Certain Di¤erence-Di¤erential Equation," Journal of the London Mathmatical Society, 25 (1950), 226-232.
[3] Invernizzi, S., and A. Medio, "On Lags and Chaos in Economic Dynamic
Models," Journal of Mathematical Economics, 20 (1991), 521-550.
[4] Matsumoto, A., "Note on Goodwin’s 1951 Nonlinear Accelerator Model withan Investment Lag," Journal of Economic Dynamics and Control, 33(2009), 832-842.
[5] Matsumoto, A., and F. Szidarovszky, "Delay Di¤erential Neoclassical
Growth Model," Journal of Economic Behavior and Organization, 78(2011a), 272-289.
[6] Matsumoto, A., and F. Szidarovszky, "An Elementary Study of a Class of
Dynamic Systems with Single Time Delay," DP 161 (http://www2.chuou.ac.jp/keizaiken/discuss.htm), Institute of Economic Research, Chuo
University, (2011b).
[7] Naimzada, A., "A Delay Economic Model with Flexible Time Lag," mimeo,
2011.
[8] Puu, T. Attractors, Bifurcations and Chaos, 2003, Berlin/Heidelberg/New
York, Springer-Verlag.
[9] Wright, E. M., "On a Class of Prey-Predator Population Models with Time
Lag," Acta Mathematica Applicatae Sinica, 11 (1955), 12-21.

17