IERCU
Institute of Economic Research, Chuo University
50th Anniversary Special Issues
Discussion Paper No.215

Dynamic Monopoly with Demand Delay

Akio Matsumoto
Chuo University

Ferenc Szidarovszky
University of Pécs

January 2014I

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, JapanDynamic Monopoly with Demand Delay
Akio Matsumotoy
Chuo University

Ferenc Szidarovszkyz
University of Pécs

Abstract
This study analyses the dynamics of nonlinear monopoly. To this end,
the conventional assumptions in the text-book monopoly are modi…ed;…rst, the complete information on the market is replaced with the partial
information; second, the instantaneous information is substituted by thedelay information. As a result, since such a monopoly is unable to jump,with one shot, to the optimal point for which the pro…t is maximized,
the monopoly has to search for it. In a continuous-time framework, thedelay destabilizes the otherwise stable monopoly model and generates
cyclic oscillations via a Hopf bifurcation. In a discrete-time framework,
the steady state bifurcates to a bounded oscillation via a Neimark-Sacker
bifurcation. Although this has been only an introduction of delay into the
traditional monopoly model, it is clear that the delay can be a source ofessentially di¤erent behavior from those of the nondelay model.

The authors highly appreciate the …nancial supports from the MEXT-Supported Program
for the Strategic Research Foundation at Private Universities 2013-2017, the Japan Society for
the Promotion of Science (Grant-in-Aid for Scientifc Research (C) 24530201 and 25380238)
and Chuo University (Grant for Special Research). The usual disclaimers apply.
y Professor, Department of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji,
Tokyo, 192-0393, akiom@tamacc.chuo-u.ac.jp
z Professor, Department of Applied Mathematics, University of Pécs, Ifúság, u.6, H-7624,
Pécs, Hungary. szidarka@gmail.com

1

1

Introduction

Implicit in the text-book monopoly is an assumption of complete and instantaneous information or knowledge available to economic agents at free of charge.
Under such circumstances, knowing the certain price and cost functions, themonopolist can make an optimal decision of price and output to maximize its
pro…t and realize it. As a result, the text-book monopoly model becomes staticin nature. There are, however, many empirical works to indicate that such anassumption of rational economic agents goes too far. In reality the monopolist
is boundedly rational and adjusts its price and output as a function of its limited knowledge and past experiences. To …ll this gap, we propose, in this study,
to relax this assumption and develop a dynamic monopoly model. In particular, we assume …rst that the monopolist has only partial information about themarket condition and second that the monopolist obtains it with time delay. In
natural consequence of these alternations, the monopolist cannot jump to the
optimal point but searches for it with using the actual data obtained throughthe market. The modi…ed model becomes dynamic in nature. This is the issue
far outside the scope of the text book monopoly and it is what we will consider
in this study.
In the recent literature, various learning processes of the boundedly rationalmonopolist have been extensively studied. Puu (1995) constructs a discrete-time
monopoly model in which price function is cubic and cost function is linear. It
is shown that the gradient learning or search process based on locally obtained
information might behave in an erratic way under the condition that the pricefunction has an in‡ection point. Assuming that the monopolist uses a ruleof thumb to determine quantity to produce, Naimzada and Ricchiute (2008)
reconsider Puu’s model with a linear cost function and a cubic price function
without the in‡ection point. Their model is then generalized by Askar (2013)
who replaces the cubic function with higher-order polynomials. Matsumoto andSzidarovszky (2013) further generalize Asker’s model by introducing the moregeneral type of the cost function. Since those models are described by one
dimensional di¤erence equation, chaotic dynamics can arise via period-doublingbifurcation.
In this study we reconsider a dynamic monopoly model from two di¤erent
points of view. First, to detect the e¤ect caused by non-instantaneous information, the dynamic process is constructed in continuous-time scales and a …xed
time delay is introduced. Second, we discretize the continuous process to obtaina "delay" discrete process and analyse the delay e¤ect on discrete dynamics. In
both models, local stability of a stationary state is analytically considered andglobal dynamics is numerically examined.
The paper is organized as follows. In Section 2, the delay di¤erential modelis presented and stability switch is considered. In Section 3, the delay di¤erence
model is constructed to give rise to the emergence of Neimark-Sacker bifurcation.
And …nally, Section 4 concludes the paper.

2

2

Delay Di¤erential Dynamics

Consider a single product monopoly that sells its product to a homogeneous
market. Let q denote the output of the …rm, p(q) = a bq the price function
and C(q) = cq the cost function.1 Since p(0) = a and j@p(q)=@qj = b; we call a
the maximum price and b the marginal price. There are many ways to introduce
uncertainty into this framework by considering a; b or c uncertain. In this study,
it is assumed that the …rm knows the marginal price and the marginal cost but
does not know the maximum price. In consequence it has only an estimate ae (t)of it at each time period. So the …rm believes that its pro…t is
e

= (ae

bq)q

cq;

its best response isae c2band the …rm expects the market price to beqe =pe = aebq e =

ae + c:2(1)

However, the actual market price is determined by the real price function
pa = abq e =

ae + c
:
2

2a

(2)

Using these price data, the …rm updates its estimate. The simplest way foradjusting the estimate is the following. If the actual price is higher than the
expected price, then the …rm shifts its believed price function by increasing thevalue of ae ; and if the actual price is the smaller, then the …rm decreases thevalue of ae : If the two prices are the same, then the …rm wants to keep its
correct estimate of the maximum price. This adjustment or learning process
can be modeled by the following di¤erential equation:a_ e (t) = (ae (t)) [pa (t)pe (t)] ;where (ae ) > 0 is the speed of adjustment. Substituting relations (1) and (2)gives the adjustment equation as a di¤erential equation with respect to ae :a_ e (t) = (ae ) [aae (t)] :(3)

For analytical simplicity, we assume that(ae ) = kae ; k > 0;
1 Linear functions are assumed only for the sake of simplicity. We can obtain a similar
learning process to be de…ned even if both functions are nonlinear. It is also assumed forthe sake of simplicity that the …rm has perfect knowledge of production technology (i.e., costfunction).

3

so equation (3) is reduced to the logistic equation,a_ e (t) = kae (t) [aae (t)](4)

which is a nonlinear di¤erential equation. Notice that equation (4) has twosteady states, ae = 0 and ae = a: Small perturbation from ae = 0 satis…es
the linear equation a_ e (t) = akae (t); which shows that ae = 0 is unstable withexponential growth. We thus only need to consider the stability of the positive
steady state ae = a: The steady state corresponds to the true value of the
maximum price.
If there is a time delay in the estimated price, then equation (4) has to bemodi…ed asa_ e (t) = kae (t) [a ae (t)] :(5)
By introducing the new variable z(t) = ae (t) a; the linearized version ofequation (5) becomesz(t)_ + z(t
)=0
(6)
where
= ak. As a benchmark for stability analysis, we start with the nodelay case. If there is no delay, = 0, then equation (6) becomes an ordinarydi¤erential equation with characteristic polynomial + : So the only eigenvalue
is negative implying the local asymptotic stability. If > 0; then the exponential
form z(t) = e t u of the solution reduces the characteristic equation to thefollowing form:
+ e
= 0:
(7)
This is a transcendental equation. Notice that the only eigenvalue is negativewhen= 0. Notice also that
= 0 is not a solution of equation (7). Forsu¢ ciently small deviation of
from zero, the real parts of the eigenvalues
are still negative by continuity. We seek conditions of such that the realparts change from negative to positive. Since stability is changed to instability
under this condition, it is often called stability switch. At this critical value of; the characteristic equation must have a pair of purely imaginary eigenvalues,= i : If is an eigenvalue, then its complex conjugate is also an eigenvalue.
So, without loss of generality, we can assume that > 0: So equation (7) canbe written as
i + e i = 0:
Separating the real and imaginary parts, we obtain
cos

=0

and
sin

= 0:

Therefore
cos
implying that=

= 0 and sin=

leading to in…nitely many solutions,
=

1
2

+ 2n

for n = 0; 1; 2; :::4(8)

The solution

with n = 0 forms a downward-sloping curve with respect to ,
=

with

2

= ak:

Applying the main theorem in Hayes (1950) or the same result obtained differently in Szidarovszky and Matsumoto (2012), we can …nd that this curve
divides the non-negative ( ; ) plane into two subregions; the real parts of theroots of the characteristic equation are all negative in the region below the curveand for some roots are positive in the region above. This curve is often called
the partition curve separating the stability region from the instability region.
Notice that the critical value of decreases with ; so a larger value of causedby the high speed of adjustment and/or the larger maximum price makes thesteady state less stable.
We can easily prove that all pure complex roots of equation (7) are single.
If is a multiple eigenvalue, then it must solve equations
+ e

=0

and
1+ e

(

) = 0:

Based on the …rst equation, the second equation becomes
1+

=0

or

1

=

implying that becomes a real negative value which contradicts the assumptionthat it is purely imaginary.
In order to detect stability switches and the emergence of Hopf bifurcation,
we select as the bifurcation parameter and consider as function of ; =( ). By implicitly di¤erentiating equation (7) with respect to , we have
d
+ e
d

d
d

=0

implying that
d
=
d
With

2

:

1+

=i ;
Re

d
d

2

= Re

1+i
2

=

> 0:
1 + ( )2
So the sign of the real part of an eigenvalue changes from negative to positiveand it is a Hopf bifurcation point of the nonlinear learning process (5) with one
delay. Thus we have the following result.
5

Theorem 1 For the logistic adjustment process (5), the steady state is locallyasymptotically stable if <and locally unstable if < s: : Hopf bifurcation
occurs if =and a stable limit cycle exist for >
where
=

2

and

= ak:

The delay logitstic adjustment process can have periodic solutions for a large
range of value of ; the product of the maximum price a and the adjustmentcoe¢ cient k. The period of the solution at the critical delay value is 0 =2 = ; which is 4 0 :An intuitive reason why stability switch occurs only at the critical value ofwith n = 0 is the following. Notice …rst that the delay di¤erential equation
has in…nitely many eigenvalues and second that their real parts are all negative
for < . When increasing arrives at the partition curve, then the realpart of one eigenvalue becomes zero and its derivative with respect toispositive implying that the real part changes its sign to positive from negative.
Hence the steady state loses stability at this critical value. Further increasing
crosses the ( ; ) curve de…ned by equation (8) with n = 1 where the real partof another eigenvalue changes its sign to positive from negative. Repeating thesame arguments, we see that at each intersection one more eigenvalue changes its
real part from negative to positive, so stability cannot be regained and therefore
no stability switch occurs for any n 1. Hence stability is changed only whencrosses the partition curve.
Theorem 1 is numerically con…rmed. Given a = 2; a bifurcation diagramwith respect to is depicted in Figure 1(A). It is seen that the steady state
loses stability at =and bifurcates to a cyclic oscillation for > . In
addition, given = a ; Figure 1(A) indicates the maximum and the minimum
values of the trajectory are denoted by yM and ym : Figure 1(B) illustrates alimit cycle having the same extremum in the phase plane.

(A) Bifurcation Diagram(B) Limit cycle with

Figure 1. Cyclic oscillations for
6

>

=

a

We can discuss another delay adjustment process that is a hybrid of equation(4) and equation (5),a_ e (t) = kae (t) [a(!ae (t) + (1!)ae (t))](9)

where ! is a positive constant less than unity. It can be seen that equation (9) is
reduced to equation (4) when ! goes to unity and to equation (5) when ! goesto zero. The steady state of equation (9) is equal to the maximum price andthus the same as the one of equation (4) as well as equation (5). The linearized
equtaion becomesz(t)_ + !z(t) + (1

!)z(t

)=0

and its characteristic equation is+ ! + (1

!)e

= 0:

Using the similar arguments, we can obtain the results including that the delay
becomes harmless if the instanteneous term ae (t) is dominant in equation (9):Theorem 2 (i) If !
1=2; then the steady state of the hybrid logistic adjustment process (9) is locally asymptotically stable for all delay0; (ii) if! < 1=2; then the steady state is locally asymptotically stable if < , loses
stability for=and bifurcates to a limit cycle via a Hopf bifurcation if
>
where
p
1 2!
1
1
sin
:
= p
1 !
ak 1 2!

3

Delay Discrete Dynamics

Our concern in this section is on how the di¤erent choice of the time scale a¤ects
dynamics examined in the previous section. Toward this end, we discretize thedelay di¤erential equation (4) by replacing a_ e (t) with ae (t + 1) ae (t) to obtainae (t + 1) = ae (t) + kae (t)(a

ae (t

))

(10)

and then reconsider local and global dynamics in discrete time. The positive
steady state of equation (5) remains as a steady state of this di¤erece equation.
We mention that this discrete-time equation has a -step delay when
1.2
The remaining of this section starts with the case of = 0 and then, proceed
the cases of
1 in detail to concentrate on delay e¤ects in the discrete-timeframework.
If = 0; then equation (10) becomes a nonlinear …rst-order di¤erence equationx(t + 1) = x(t) + kx(t)(a x(t))(11)
2 A sailent feature of a discrete-time equation is that the equation involves at least onedi¤erence or time-delay of the dependent variable. So we refere to the -step delay when is
greater than unity.

7

where we introduce the new variable x = ae : Changing the variable again byxz = 1+akk

reveals that equation (11) can be reduced to the familiar form,z(t + 1) = (1 + ak)z(t)(1z(t)):

It is now well known that the logistic equation can generate wide variety ofdynamics ranging from a periodic cycle to chaos according to the speci…cationof the coe¢ cient 1 + ak if the steady state is locally unstable.
If = 1; then equation (10) has one-step delay and then becomes a nonlinear
second-order di¤erence equationx(t + 1) = x(t) + kx(t)(a

x(t

1))

(12)

which can be converted to an equivalent 2D system of …rst-order di¤erence
equations,y(t + 1) = x(t);(13)
x(t + 1) = (1 + ak)x(t) kx(t)y(t):The linearized system around the steady state x = y = a is
x (t + 1)
y (t + 1)

=

1
1

ak
0

x (t)
y (t)

where the subscript implies that the variable with this subscript is the difference between its value and the steady state. The characteristic equation istransformed into a quadratic equation,2+ ak = 0:

The following three conditions imply that the quadratic polynomial
a2 has roots inside the unit cycle,(14)2+ a1 +

1 + a1 + a2 > 0;1

a1 + a2 > 0;

1

a2 > 0

(15)

where
a1 =

1 and a2 = ak.

The …rst and second conditions of (15) are always satis…ed and so is the third
condition if and only if
ak < 1:

8

Taking a = 2 and selecting k as the bifurcation parameter, we illustrate thebifurcation diagram in Figure 2(A) in which stability of the steady state ischanged to instability at k s = 1=a (= 0:5) and cyclic behavior emerges fork > k s : When k arrives at k ' 0:635; the non-negativity condition is violated
resulting in the birth of economically uninteresting behavior.
We further extend our analysis to a two-step delay (i.e., = 2) where themarginal revenue includes the delayed information obtained at period t 2: The
dynamic equation (10) is now a third-order di¤erence equation,x(t + 1) = x(t) + kx(t)(a

x(t

2)):

(16)

This can be written as a 3D system of …rst-order di¤erence equationsy(t + 1) = x(t)z(t + 1) = y(t)(17)

x(t + 1) = (1 + ak)x(t)kx(t)z(t);where the steady state is (x; y; z) with x = y = z = a: Linear approximation ofequations (17) yields the linearized system having the form
1 0
10
1
0
1 0
ak
x (t)
x (t + 1)@ y (t + 1) A = @ 1 0
0 A @ y (t) A
(18)
0 1
0
z (t)
z (t + 1)
and the corresponding characteristic equation is cubic,
3

2+ ak = 0:(19)

The steady state is locally asymptotically stable if all eigenvalues of equation(19) are less than unity in absolute value. Farebrother (1973) has proved thatthe most simpli…ed form of the su¢ cient and necessary conditions for the cubic
equation 3 + a1 2 + a2 + a3 to have roots only inside the unit cycle are
1 + a1 + a2 + a3 > 0;1

a1 + a21

a2 + a1 a3

a3 > 0;(20)
a23 > 0;a2 < 3wherea1 =1; a2 = 0 and a3 = ak:

It can be veri…ed that the …rst and fourth conditions are always satis…ed while
the second and third condition holds ifp
5 1' 0:618:ak <2
9

The bifurcation diagram with a = 1 is illustrated in Figure 2(B) where k s '0:618. It can be seen that Hopf bifurcation emerges for k > k s :3
If = 3; then the characteristic equation is 4
+ = 0 with = ak: For
the quartic equation4+ a1 3 + a2 2 + a3 + a4 = 0

the su¢ cient and necessary condition that all roots are inside the unit circle are(see Farebrother, 1973) as follows:
1a4 > 03 + 3a4 > a2
1 + a1 + a2 + a3 + a4 > 0
1(1

a1 + a2(a4 )2 )a4 )(1a2 (1a3 + a4 > 0a4 )2 + (a1a3 )(a3

a1 a4 ) > 0

To our case, a1 = 1; a2 = a3 = 0 and a4 = . The …rst four conditions areclearly satis…ed if < 1 and the last condition can be reduced to the following:f( ) =

32

2 + 1 > 0:

Clearlyf ( 1) = 1; f (0) = 1; f (1) =Since f 0 ( ) = 3 2
1 =

2
1

1; f ( 1) =1 and f (1) = 1:

2 having two roots
p
3

7

'

p
1+ 7
0:548 and 2 =' 1:215;3f ( ) increases in intervals ( 1; 1 ) and ( 2 ; 1); and decreases in ( 1 ; 2 ).
Notice that f ( 1 ) > 0 and f ( 2 ) < 0; so f ( ) has three real roots: one is
negative, two positive in intervals (0; 1) and ( 2 ; 1). Since< 1 and thesmallest positive root is approximately 0:445;3 the stability condition is k <0:445: It is numerically con…rmed that the steady state is violated via NeimarkSacker bifurcation for k > 0:445: Although the critical values k s seem to decreaseas the value of increases,
k s = 1 if

= 1; k s ' 0:618 if= 2 and k s ' 0:445 if= 3;

3 With Mathematica, it can be found that the critical value has the following form,

1
k =
3
s

3

p
72=3 (1 + i 3)p
22=3 ( 1 + i3 3)1=3p(1 + i 3)

1=3p
1 + i3 3)

7
(
2

6

10

' 0:445this fact has not been analytically con…rmed yet.
Applying the same argument to the case of the general case, we have theadjustment process described by a ( + 1) th-order di¤erence equation,x(t + 1) = x(t) + kx(t)(a

x(t

))

and a ( + 1) th-order characteristic equation,+1+ ak = 0:

For a larger value of
4; we do not have the simplifed stability condition but
the Samuelson or the Cohn-Schur conditions for the n-th order equation canbe applied to determine the critical value of k for the birth of Neimark-Sacker
bifurcation.4 We summarize the main result on the delay di¤erence adjustment
process:
Theorem 3 Given the maximum price, the discrete-time adjustment process(10) has the critical value of the adjustment coe¢ cient k s and the steady stateis locally asymptotically stable if k < k s , loses stability for k = k s and bifurcatesto a limit cycle via Neimark-Sacker bifurcation if k > k s .

(A)= 1 and a = 2(B)= 2 and a = 1

Figure 2. Bifurcation diagrams with di¤erent steps

4

Concluding Remarks

In this study, we analyzed the delay dynamics of a nonlinear monopoly. Twoconventional assumptions in the traditional monopoly model are modi…ed: the4 The forms of both conditions are found in Gandolfo (2009). In either from, the stability
condition becomes complicated as the order of the equation increases.

11

information obtained from the market are assumed to be limited and delayed.
As a natural consequence, the monopoly is unable to jump, with one shot, tothe optimal point but revises its decision by taking transaction data experiences
obtained from the market into account. In either the continuous-time framework or the discrite-time framwork, the steady state is locally asymptoticallystable for the smaller values of delay and bifurcates to a limit cycle via a Hopf
bifurcation in the continuous-time framework and via a Neimark-Sacker bifurcation in the discrete-time framework for the larger values. Delay monopoly
generates very di¤erent dynamics of the those of the text-book monopoly.

References[1] Askar, S. S. (2013), "On Complex Dynamics of Monopoly Market," Economic Modeling, 31, 586-589.
[2] Farebrother, R. W. (1973), "Simpli…ed Samuelson Conditions for Cubic
and Quartic Equations," The Manchester School of Economic and Social
Studies, 41(4), 396-406.
[3] Gandolfo,
G.,(2009),
Economic
Berlin/Heidelberg/New York, Springer

Dyamics,

4th

Edition,[4] Hayes, N. D. (1950), "Roots of the Transcendental Equations Associatedwith Certain Di¤erence-Di¤erential Equation," Journal of London Mathematical Society, 25, 226-232.
[5] Matsumoto, A. and F. Szidarovszky (2013), "Complex Dynamics of Monopolies with Gradient Adjustment," IERCU DP # 209, Institute of
Economic Research, Chuo University.
[6] Matsumoto, A. and F. Szidarovszky (2012), "An Elementary Study of aClass of Dynamic System with Single Time Delay," IERCU #161, Institute of Econoic Research, Chuo University, forthecoming in CUBO A
Mathematical Journal.
[7] Naimzada, A. K. and G. Ricchiuti (2008), "Complex Dynamics in a
Monopoly with a Rule of Thumb," Applied Mathematics and Computation, 203, 921-925.
[8] Puu, T., (1995), "The Chaotic Monopolist," Chaos, Solitions and Fractals,
5(1), 35-44.

12