Discussion Paper No.341
Delay Dynamics in Nonlinear Monopoly
with Gradient Adjustment

Akio Matsumoto
Chuo University

Ferenc Szidarovszky
Corvinus University

January 2021

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, JapanDelay Dynamics in Nonlinear Monopoly with
Gradient Adjustment
Akio Matsumotoy

Ferenc Szidarovszkyz

Abstract
Two types of boundedly rational monopolies are considered, when they
are unable to determine the pro…t maximizing output levels. In the …rst
case the monopoly knows the price function and in the second case it canaccess only past output and price values. In applying gradient dynamics
the marginal pro…t is either known or approximated by …nite di¤erences
based on two past pro…t data. Stability conditions are derived …rst withdiscrete time scales, which are also applied in a special case. Two modelsof continuous dynamics are then introduced. The …rst is a natural modi…cation of the discrete model, and the other includes an inertia coe¢ cientwith the derivative. In each case a delay di¤erential equation is obtainedwith two delays. Stability conditions are derived and the stability switching curves are constructed and illustrated.

Keywords Gradient adjustment, Delay di¤erential equation, Boundedly rational monopoly, Discrete-time and continuous-time dynamics, Stability switching curves

The …rst author highly acknowledges the …nancial supports from the Japan Society for
the Promotion of Science (Grant-in-Aid for Scienti…c Research (C) 20K01566).
y Department of Economics, Chuo University, 742-1, Higashi-Nakano, Hachioji, Tokyo, 1920393, Japan, akiom@tamacc.chuo-u.ac.jp.
z Department of Mathematics, Corvinus University, Budapest, Fövám tér 8, 1093, Hungary,szidarka@gmail.com.

1

1

Introduction

It is well-known that a monopolist in an elementary textbook of microeconomics
is assumed to be rational in the sense that it has the perfect information on themarket and instantaneous responses to changing circumstances. Accordingly,
such a monopolist can choose the levels of price and output that maximizes its
overall pro…t and can adjust its decisions in no time without any di¢ culties ifsome exogenous changes occur. It is also well-known that the decision-makers
in the real world are boundedly rational and thus have to decide under limited
information and delayed responses. We can say this behavioral di¤erence inother words. The rational monopolist can jump to the optimal point of output
and price in one shot without any adjustments. In consequence, output as wellas price will not change over time (i.e., no dynamic consideration is necessary)
unless environmental phenomenon changes. The boundedly rational monopolist, on the other hand, can make mistakes due to limited information. It might
produce a di¤erent amount of output and set a di¤erent value of price otherthan the optimal ones. Noticing the mistake and revising the decision, it experiences time delays in collecting past data of price and output associated withuncertainty, information and implementation delays. Output (and price) willvary in every subsequent time period. The paper’s main purpose is to shed lighton such an adjustment or dynamic process of output of the boundedly rationalmonopolist.
In the existing literature, the gradient method is often adopted to describe
the adjustment process of the boundedly rational monopolist toward the pro…tmaximizing output. Accordingly, the monopolist increases the output level if
its marginal pro…t is positive, decreases if negative and maintains the same
output level if zero. Two types of models are known to introduce the method,
discrete-time models and continuous-time models. It is demonstrated that theformer could generate chaotic dynamics if the involved nonlinearities are strong
enough. Among others, we mention Puu (1995) that follows Baumol and Quandt(1964) constructing a model of monopoly with a linear cost function and a cubic
price function with in‡ection points. Naimzada and Ricchiuti (2008) replace
Puu’s price function with a cubic function having no in‡ection points. Askar(2013) assumes a general concave price function. Elsadany and Awad (2015)
introduce a log-concave function. In a continuous-time framework, Matsumoto
and Szidarovszky (2012, 2014) build a monopoly model, focusing on the e¤ects
caused by time delays and show the delay e¤ect can be a source of complex
dynamics as well as simple dynamics. In those studies, it is assumed that theform of the demand function could be known or estimated correctly by using
the history of output and price.1 In this study, we introduce two boundedly
rational monopolists, one knows the form of the price function and the otherknows only a few points on it. Further, neither of them knows the equilibrium
output value and search for it based on the gradient method. We analyticallyand numerically consider how di¤erent amounts of information or knowledge on
1 Even if the price function is known, it might be possible that a monopolist is endowedwith limited computational skills to solve the pro…t maximization problem.

2

the price function a¤ect their search behavior in discrete-time and continuoustime framework.
The rest of this paper is organized as follows. Section 2 builds a basicmodel. Section 3 considers the learning process in a discrete-time model. Section 4 constructs a continuous-time model from the discrete-time model via
Euler approximation and then considers the same subject in a continuous-timeframework. Section 5 investigates a continuous-time model with inertia. Finally,
the concluding remarks and future research directions are given in Section 6.

2

Basic Model

Consider a monopoly that produces the quantity of output x . The price andproduction cost functions are continuously di¤erentiable and are denoted, respectively, as p(x) and c(x). It is also assumed that p(x) has a unique inverse.
The pro…t function is de…ned to be(x) = p(x)xc(x).

According to the textbook monopoly theory, the interior optimal output xM
satis…es the …rst-order condition of the pro…t-maximization,
d (x)
= 0:
dx

(1)

The monopoly price pM is determined by the market-clearing condition for x,pp(x) = 0:(2)

Assuming the strict concavity of the pro…t function, the rational monopolist
determines unique equilibrium values, xM and pM = p(xM ), through (1) and(2). It is also implicitly assumed that the rational monopolist has (i) full information on the form of the price function and (ii) enough computability to solve
the two conditions, (1) and (2). Hence, the rational monopoly …rm can jump to
the equilibrium output and price with one-shot (i.e., no dynamics).
In this study, we relax either or both of information requirements (i) and (ii)and consider what happens if the monopolist sets its output at some level otherthan xM . First, we introduce two di¤erent monopolists according to how muchinformation they have. We call the monopoly the "knowledgeable monopolist"or the k-monopolist if it has requirement (i) but does not have requirement(ii)2 and the "limited monopolist" or the `-monopolist if it has neither of therequirements but possesses the values of x and p in the past two periods of time.
It is implicitly assumed that the price is determined so as to clear the market
even when x is di¤erent from the equilibrium (i.e., p = p(x) for all x). Second,
those monopolists with insu¢ cient information adjust their outputs based on
the observation of a pro…t change per a unit change of output. If the change is
2 Clower (1959) calls a monopolist "knowledgeable monopolist" under the similar circumstance.

3

positive, negative or unchanged, then the output levels are increased, decreased
or maintained at the same levels. Since the k-monopolist knows the analytic
form of its pro…t, its pro…t change can be obtained by di¤erentiating the pro…tfunction. If the change is denoted as= x; then

x

'

d (x)
:
dx

(3)

On the other hand, the `-monopolist does not know the form of the pro…tfunction but can observe its pro…ts at two di¤erent periods. The pro…t change
at time t can be described by
x

=

(x(t
x(t

1 ))
1)

(x(t
x(t

2 ))
2)

where t1 and t2 are earlier time periods with known pro…t values, so
and 2 are nonnegative integers and 1 < 2 .

3

(4)
1

Discrete-time Dynamics

If discrete time scales are assumed and the earlier time periods are selected as
close to t as possible (i.e., 1 = 1 and 2 = 2), then the k-monopolist adjusts
its output in proportion to the marginal pro…t change,x(t) = x(t1) + Kd (x(t 1))dx(t 1)(5)where this algorism is often called the gradient adjustment process. The `monopolist determines its output level, following the formulation,x(t) = x(t1) + K(x(t 1))x(t 1)(x(t 2))x(t 2)(6)that follows Puu (1995). In both equations, K > 0 is the adjustment coe¢ cient.
This section focuses on the discrete dynamics controlled by (5) and (6) andresults to be obtained are extending Naimzada and Ricchiuti (2008), Matsumoto
and Szidarovszky (2012) and Askar (2013).
The asymptotical behavior of nonlinear model (5) can be obtained by linearization around the steady state xM . At the steady state x(t) = x(t 1) = xM ;we have the linear form of (5),x(t) = x(t1) + K 00 (xM )x(t

1)

(7)

where 00 (xM ) is the second derivative evaluated at x = xM . In order to guarantee that the …rst order condition at xM provides maximum, we make thefollowing assumption,
Assumption 1.

00(xM ) < 0.
4

The eigenvalue of equation (7) is
=1

A

where
A=
Since

K 00 (xM ) > 0:(8)

< 1 always, the stability condition for the k-monopolist is

>

1 or

A < 2:For the `-monopolist,3 we have the following from (6)1) + K 0 (z):x(t) = x(twith z being between x(talso xM ; therefore

1) and x(t(9)2). At the steady state, z has to be

xM = xM + K 0 (xM );implying that xM is a stationary point of the pro…t function. Notice that
@x(t)
= 1+K
@x(t 1)0(x(t1)) [x(t

1)

x(t

(x(t

2)]
1)

[ (x(t

1))

(x(t

2))]

2

x(t

2))

where the numerator can be written as0(x(t 1)) (x(t

1)

x(t

where z is between x(t

2))+

0

1) and x(t(x(t1)) (x(t

2)

x(t

1)) +

00

(z)
(x(t
2

2). Therefore@x(t)K 00 (z)=1+:@x(t 1)
2
At the equilibrium z = xM , therefore this derivative becomes@x(t)K 00 (xM )=1+=1@x(t 1)2

A=2:

Similarly, the other derivative at the equilibrium is@x(t)K 00 (xM )==@x(t 2)2

A=2:

The linearized equation of (6) becomesx(t) = (1

A=2) x(t1)(A=2) x(t

2)

(10)

3 Notice that the `-monopolist is unable to manipulate the following calculations since itdoes not know the form of the pro…t function.

5

2)

x(t

2

1))

with characteristic equation2(1A=2) + A=2 = 0:

(11)

The steady state is locally asymptotically stable if
1(1A=2) + A=2 > 0(12)
A=2 < 1

which can be simpli…ed as 0 < A < 2:
Theorem 1 The steady state of dynamic equations (5) for the k-monopolist
and (6) for `-monopolist is locally asymptotically stable if 0 < A < 2 and locallyunstable if A > 2:
Next we adopt a general concave price function to con…rm the analyticalresult by examining numerical examples,
p=a

2 Z+

bx ;

(13)

that is used in Askar (2013). With a marginal cost c,4 the pro…t function is(x) = (pc)x = (a0cbx1+ ;c)x

the …rst derivative is(x) = a(1 + ) x :Hence the dynamic equation for the k-monopolist is obtained from (5),x(t) = x(tSolving01) + K [ac(1 + )x (t

1)]

(14)

(x) = 0 presents the equilibrium value, xM ,
xM =

a c
(1 + )b

1

:

(15)

The second derivative at the equilibrium point xM is
100(xM ) =b(1 + )a c
(1 + )b:

Apparently the pro…t function satis…es the second-order condition for pro…tmaximization due to Assumption 1. The stability condition for the k-monopolist
is, according to Theorem 1,
1

A=

K

00

(xM ) = K b(1 + )a c(1 + )b

4 It has been checked that the convex cost function c(x) = x ; 2
the general properties of the result to be obtained.

6

<2

(16)
does not a¤ectthat is reduce tobK < 1 if= 1 or the price function is linear ,3bKxM < 1 if

= 2 or the price function is quadratic,

and6bKx2M < 1 if

= 3 or the price function is cubic.

For the `-monopolist, the pro…t change is given by(x(t 1))(x(t 2)) = (a c) [x(t1)2)] b x +1 (tx(tx +1 (t1)2) :where the second bracketed term on the right side can be factorized as!
Xkk[(x(t 1) x(t 2)]x(t 1)x (t 2)k=0Hence the adjustment process is rewritten asx(t) = x(t1) + K fa

whereg [x(t1); x(t2)] =cbg [x(t1); x(t

2)]g

(17)

X

x

k

(t

1)xk (t

2):

1)

#
@g
bK
x(t 2)@x(t 2) x(t 2)=xM
(18)

k=0

The corresponding linearized equation is"!
@gx(t) =1 bKx(t@x(t 1) x(t 1)=xM
where

@g
@g
(1 + )
xM 1 :==@x(t 1) x(t 1)=xM@x(t 2) x(t 2)=xM
2The linear equation is reduced to
x(t)

(1

bK) x(t1) + bKx(t2) = 0 if

= 1;

x(t)

(1

3bKxM ) x(t1) + 3bKxM x(t2) = 0 if

=2

x(t)

1 + 6bKx2M x(t1) + 6bKx2M x(t2) = 0 if

= 3:

and
In consequence, the stability condition for the adjustment process for the `monopolist isbK < 1 if= 1; 3bKxM < 1 if

= 2 and 6bKx2M < 1 if

= 3:

Notice that the stability conditions for the k-monopolist and the `-monopolist
are identical for = 1; 2; 3: This result is con…rmed for any integer as statedin Theorem 1. In the following numerical examples, since we repeatedly use theparameter values that Naimzada and Ricchiuti (2008) take, we make it as an
assumption,
7

Assumption 2. a = 4; b = 0:6 and c = 0:5.
Figure 1(A) illustrates the two bifurcation diagrams for K with = 3 andAssumption 2. The red one is for the the k-monopolist and the blue one is for the`-monopolist. It is observed that for K < K0 = 6bKx2M ' 0:216; both systems
are locally asymptotically stable. The stability is lost for K = K0 and bifurcatesto a periodic cycle for K > K0 via a Hopf bifurcation. The red diagram has asomewhat familiar shape, and thus the adjustment process for the k-monopolist
gives rise to chaotic oscillations via a period-doubling cascade. On the other
hand, the blue diagram indicates that for K > K0 ; the adjustment process forthe `-monopolist can generate only a periodic cycle. Further, its lower branch
becomes negative for a larger value of K; losing economic meanings. It is worth
mentioning that the `-monopolist can arrive at the steady state if it adjusts
its output level cautiously with a small value of the adjustment coe¢ cient,although it has limited information only on prices and output levels in the lasttwo periods. In Figure 1(B), the phase diagram for the `-monopolist is plottedfor K1 = 0:28 and shows that there is a period-4 cycle. After 5000 iterations ofequation (17), the ordinates of points A and C take very close values, 0:937 and
0:938. As a result, it seems that the blue diagram has only three branches for
K > K0 in Figure 1(A). However, the middle branch actually consists of two
similar curves. The vertical dotted line at K = 0:28 crosses the blue diagram
four times, although it looks like crossing three times.

(A) Bifurcation diagrams(B) Period-4 cycle

Figure 1. Dynamics generated by (14) and (17)
We now extend our analysis to the case in which only the mth-latest information ( 1 = m) and the nth-latest information ( 2 = n) are available forn > m > 0 and m and n being coprime intergers. In consequence, the output
adjustment process is
x(t) = x(tn) + K(x(t m))x(t m)
8(x(t n))x(t n)(19)

where xM is the steady state. Its linear version is an nth-order di¤erence equation,x(t) = (1 A=2) x(t m) + ( A=2) x(t n):
If 0 < A < 2; thenj1

A=2j + j A=2j = 1:µUnder this special condition, according to Corollary 3.1 of Cermák
and Jánský
(2015), the steady state of (19) is asymptotically stable if(1m

A=2)n ( A=2)

< 0 for any m and n:(20)

Since 1 A=2 > 0 and A=2 < 0; it is clear that the inequality holds if m isodd. Therefore, when we take m = 1, the stability condition for the higher-orderdi¤erence equation of (19) is given as follows:
Theorem 2 If 0 < A < 2; then the steady state of the nth-order di¤ erence
equation of (19) is locally asymptotically stable for m = 1 and any n 2;
Equation (19) with m = 1 and n = 3 is cubic,
x(t) = x(twhere g [x(tx3 (t1); x(t1) + K facbg [x(t1); x(t

3)]g

(21)

3)] has the form,

1) + x2 (t1)x(t3) + x(t1)x2 (t3) + x3 (t3):

Applying the traditional necessary and su¢ cient stability conditions shown by
Farebrother (1973) leads to 0 < A < 2 that con…rms the stability condition of(20). Under Assumption 2, Figure 2(A) presents the bifurcation diagram withrespect to K 2 [K0 0:02; 0:29], showing the birth of a periodic-cyle after K0
and complicated dynamics for K close to 0:29: However, the lower part could be
negative with no economic meaning. To see the behavior in the neighborhood ofK0 , we enlarge the diagram in a small interval [KS ; KE ] with KS = K0 0:002
and KE = K0 + 0:002: It is observed that equation (21) gives rise to a period-6
cycle just after losing stability. As the value of K increases, the upper two,the middle two and the lower two branches are getting closer, giving the wrong
image that they merge to one. After 5000 iterations of equation (19), Figure2(C) depicts a phase portrait for K1 = 0:26 that is still a period-6 cycle where
the ordinates of the higher, middle and lower points are, respectively, the same.
Returning to Figure 2(A), the bifurcation diagram indicates that it generates a9period-3 cycle after K0 but in fact, it is a period-6 cycle.

(A) Bifurcation(2) Enlargement(C) Phase portrate

Figure 2. Dynamics of the 3rd-order di¤erence equation

4

Continuous-time Dynamics I

Introducing time delays, we transform the discrete-time models to continuoustime models through Euler approximation (i.e., x(t)_' x(t) x(t 1)). For the
k-monopolist, equation (5) is modi…ed as
x(t)
_
=K

d (x(t
dx(t

))
)

(22)

and equation (6) for the `-monopolist is converted to
x(t)
_
=K

(x(t
x(t

1 ))

(x(t
x(t

1)

2 ))
2)

(23)

where > 0; 1 > 0 and 2 > 0 with 0
1 < 2 denote time delays incontinuous-time scales. The steady state of these equations are the same asthat in the di¤erence equations (5) and (6).
We …rst consider the k-monopolist case in which a linear version of delaydi¤erential equation (22) is
x(t)
_
=

Ax(t

)

(24)

where A is de…ned in equation (8). Substituting an exponential solution x(t) =e t u into (24) presents a characteristic equation,
+ Ae
Since
tion,

t

= 0:

(25)

= 0 does not solve equation (25), we suppose a pure imaginary solu= i!, ! > 0 that separates the characteristic equation to the real and10

imaginary parts,
A cos ! = 0;!

A sin ! = 0:

The …rst equation implies cos ! = 0 and the second sin ! = !=A = 1. Hencethe critical values of delay are
m =

1
!

2

for m 2 Z+ and ! = A:

+ 2m

(26)

It is apparent from (24) that the steady state is locally asymptotically stable
for = 0: The results are summarized as follows:
Theorem 3 The steady state of delay di¤ erence equation (22) is locally asymptotically stable for < 0 ; loses its stability at = 0 and then is replaced withperiodic oscillations via a Hopf bifurcation for > 0 where
0 =

2A

> 0:

We now draw attention to equation (23), the delay output adjustment equation of the `-monopolist. Similar to the discrete case, at the steady state,and@ x(t)_
K 00 (x )
=
=
@x(t
2
1)

A
2

@ x(t)_
K 00 (x )
=
=
@x(t
2
2)

A
:
2

Hence the linearized model has the form,
x(t)
_
=

A
x(t
2

1)

A
x(t
2

2 ):

(27)

with characteristic equation+
Case 0:

1 = 0;

A
e
2

1

+

A
e
2

2

= 0:

2 = 0:

Consider …rst the no-delay case. The characteristic equation (28) with
2 = 0 is=A:

The steady state is locally asymptotically stable since A > 0.
Case 1:

1 = 0;

(28)

2 >0

11

1 =

Consider now the special case of 1 = 0. Then equation (28) is reduced to a
one-delay equation,
A A
2
+ + e
= 0:
(29)
2
2
As is already seen, the steady state is locally asymptotically stable at 2 = 0. As
the value of 2 increases, stability might be lost, when = i! (! > 0). Assuming
positive value of ! does not restrict generality, since if is an eigenvalue, thenits complex conjugate is also an eigenvalue. Substituting this value of into
equation (29), we have
i! +

A A
+ (cos ! 2
2
2

i sin ! 2 ) = 0.

The separation of the real and imaginary parts gives
A + A cos ! 2 = 0;2!

A sin ! 2 = 0:

The …rst equation implies that cos ! 2 = 1; so sin ! 2 = 0 which contradicts
the second equation. Therefore there is no stability switch.5
Theorem 4 If
all 2 > 0.
Case 2:

1 = 0; then the steady state is locally asymptotically stable with

1 > 0;

2 >0

In the general case of

1 > 0 andP0 ( ) + P1 ( )e2 > 0, equation (28) can be rewritten as1+ P2 ( )e

2

=0

(30)

with
P0 ( ) =and P1 ( ) = P2 ( ) = A=2:

Before looking for stability switching curves, the following conditions should beveri…ed (Gu et al. (2005)):(i) deg[P0 ( )]max fdeg [P1 ( )] ; deg [P2 ( )]g :(ii) P0 (0) + P1 (0) + P2 (0) 6= 0:(iii) The polynomials P0 ( ); P1 ( ) and P2 ( ) do not have common roots.
(iv) lim !1P1 ( )
P2 ( )+
P0 ( )
P0 ( )< 1:

5 Mathematically, we have the same result ifcase is assumed away by assumption 1 < 2 .

12

1 > 0 and2 = 0: However this symmetricEquation (30) satis…es these conditions. Since the followings hold,deg [P0 ( )] = 1 and deg [P1 ( )] = deg [P2 ( )] = 0;
condition (i) is satis…ed. Condition (ii) is satis…ed as P0 (0) + P1 (0) + P2 (0) =A 6= 0: Condition (iii) is apparently satis…ed as P0 ( ); P1 ( ) and P2 ( ) have nocommon roots. Condition (iv) also holds, since
P2 ( )
P1 ( )+
P0 ( )
P0 ( )=

A! 0 asj j! 1:

Dividing equation (30) by , we get
1 + a1 ( )e1

+ a2 ( )e

2

= 0;

(31)

where new functions area1 ( ) =AA
and a2 ( ) =:2
2

We examine the stability switches of the non-trivial solution of dynamic equation(27) as the delays ( 1 ; 2 ) vary. The modi…ed characteristic equation (31) musthave a pair of pure conjugate imaginary roots and stability switch occurs forthe corresponding critical delays. So let = i!; ! > 0 and substitute it into
equation (31),
1 + a1 (i!)e i! 1 + a2 (i!)e i! 2 = 0
(32)
where

A
:
2!
We now solve equation (32). To this purpose, we treat the three terms in theleft hand side of equation (32) as three vectors in the complex plane with themagnitudes, 1; ja1 (i!)j and ja2 (i!)j where the absolute values are
a1 (i!) = a2 (i!) =i

ja1 (i!)j = ja2 (i!)j =A:
2!

The right hand side of equation (32) is zero, implying that if we put these
vectors head to tail, then they form a triangle as illustrated in Figure 3. Similar
triangle can be formed under the real axis. Since the sum of lengths of any
two line segments is not shorter than that of the remaining line segment in atriangle, these absolute values satisfy the following inequality conditions1

ja1 (i!)j + ja2 (i!)j ;1

ja1 (i!)j

(33)

and

13

ja2 (i!)j1(34)

Figure 3. Triangle representation of equation (32)
Relation (34) is clearly satis…ed, and (33) requires thatA!

1 or 0 < !

A:

(35)

By using the cosine rule,
2

1 =

2

= cos

1

= cos 12

1 + ja1 (i!)jja2 (i!)j2 ja1 (i!)j

!

(36)

!
:
A

Notice also that
1 =

2 2

0;

2

and
arg[a1 (i!)] = arg [a2 (i!)] = arg

i

A
3
=
:
2!
2

Hence the angle balance relations at points (0; 0) and (1; 0) imply that thestability switching curves are given as
1;m (!) =

1 3
+ (2m
! 2

1)

1 (!)

(37)

2;n (!; ) =

1 3
+ (2n
! 2

1)

2 (!)

(38)

and

where both 1 and 2 are positive with all nonnegative integer values of m andn: The results obtained are summarized in the following:
14

Theorem 5 The stability switching curve consists of the following segments,
1
SWm;n
(!) =

+
1;m (!);

2;n (!; )

2
SWm;n
(!) =

1;m (!);+
2;n (!; )! 2 [0; A]; (m; n) 2 Z ;! 2 [0; A]; (m; n) 2 Z :Using numerical example with Assumption 2, we visualize the analyticalresults. With the concave price function (13) with = 3, the dynamic equation(30) of the k-monopolist is modi…ed as
x(t)
_
=K a

4bx3 (t

c

)

(39)

and its linear version is
x(t)
_
=

Ax(t

):

Suppose that the linear equation has an exponential solution e t and = i!
with ! > 0: Then the linear equation is separated to the real and imaginaryparts,
A cos ! = 0;!

A sin ! = 0:

The …rst equation implies cos ! = 0 and the second sin ! = !=A = 1. Hencethe critical values of delay arem (!) =

1
2!

2

for m 2 Z+ and ! = A:

+ 2m

The output dynamic equation for the `-monopolist isx(t)_= K fawhere g [x(tx3 (t1 ); x(t21 ) + x (tcbg [x(t1 ); x(t2 )]g21 )x (t32 ) + x (t(40)

2 )] has the form,
1 )x(t2 ) + x(t2 ):Notice that xM is the steady state of equation (40). Some stability switching
curves are illustrated in Figure 4(A) for K = 0:2. The feasible region satisfying
2
1 is in yellow and divided into two subregions by the black segment1SW0;0(!). The steady state is stable to the left and unstable to the right. The
2
red segment in the white region is the loci of SW0;0(!): Any other segments
are located outside of Figure 4(A). Three bifurcation diagrams are illustrated in
Figure 4(B) in which x(t) is plotted against the parameter 1 . For the left-most0blue diagram, 2 is …xed at 2 and 1 is increased from 0 to a1 = 1. In Figure
4(A), the horizontal line at 2 = 2 crosses the black stability curve at point a
where its abscissa is a1 ' 0:589:6 After a1 , the steady state is no longer stable
and replaced with a limit cycle whose amplitude gets larger as 1 is increased.
As a natural consequence, the lower branch of the cycle becomes negative and6 From(37) and (38), solving 2 =+
2;0 (!)

1;0 (! a ) ' 0:589:

15

gices ! a ' 1:214 that is substituted into

loses its economic meaning. For the middle bifurcation diagram, the …xed value0of 2 is decreased to 1 from 2 and 1 is increased to b1 = 1. A thresholdbvalue is changed to 1 ' 0:743: The dynamic equation (39) gives rise to the00
red bifurcation diagram for 2 [0; a1 ] with a1 = 1:2 and a threshold valuec
1 ' 0:848: By the formulations of the pro…t change per a unit output changein (22) and (23), we have
lim

1! 2

(x(t
x(t

1 ))
1)

(x(t
x(t

2 ))
2)

=

d (x(t
dx(t

))
for
)

=

1 =

2.

It is also con…rmed that equation (40) is reduced to equation (39) when 1 =2 = : Hence the k-monopolist’s behavior can be described by increasing thevalue of 1 along the diagonal in Figure 4(A). The middle blue diagram arrives
at the lower and upper branches of the red diagram at 1 = 1 = 2 at which the
dynamic equation of the `-monopolist is identical with the dynamic equation of
k-monopolist.

(A) Feasible stable region(B) Bifurcation diagrams

Figure 4. Dynamics generated by (39) and (40)5Continuous-time Dynamics II

In this section, for analytical simplicity (at the expense of generality), we impose
Assumption 2 and assume a concave price function from the beginning,
p=a

bx3 :

The output adjustment equation for the k-monopolist in discrete-time scaleswith the delay length isx(t) = x(t

)+K a
16

c

4bx3 (t) :(41)

There are many ways to transform a discrete-time model to a continuous-timemodel. Euler approximation is one of them. In this section, applying Berezowski transformation,7 we convert equation (41) to the following continuoustime equation,x(t)_ + x(t) = x(t)+K a4bx3 (t

c

)

(42)

where is the inertia coe¢ cient. It is clearly seen that for = 0; equation (42)reduces identically to the original discrete-time equation, (41). The steady statexM in (41) is also a steady state of the new equation.
To proceed to the stability consideration, we linearize equation (42) in theneighborhood of xM to obtainx(t)_ + x(t)where A =(1A) x(t)=0(43)

K 00 (xM ) is given by
A = 12bKx2M > 0:

The corresponding characteristic equation for x(t) = e t u is
+1

(1

A) e

= 0:

When is pure imaginary (i.e., = i!; ! > 0), the characteristic equation canbe separated into the real and imaginary parts,
1(1A) cos ! = 0;

! + (1A) sin ! = 0(44)

Moving the constants of both equations in (44) to the right side, squaring them,adding them and then solving the resultant equation for ! give a positive solution,p(A 2)A> 0 for A > 2:!+ =Hence if A 2; then no stability switch occurs, otherwise, the stability mightswitch for the critical delay value,
m =

1
cos 1
!+

1
1

+ 2m

A

for m = 0; 1; 2; :::(45)

The stability of xM in (42) is summarized as follows:
Theorem 6 The steady state of equation (42) with A > 2 is locally stable for< 0 , loses stability for = 0 and bifurcates to a oscillatory cycle via a Hopf
bifurcation for > 0 where from (45) with m = 0;
0 = p

(A

2)A

cos 1

7 See Berezowski (2001) for more details.

17

1
1

A

.

With additional speci…cation of K = 0:3 and = 0:2, the bifurcation diagram given in Figure 5 con…rms the analytical results in Theorem 6. Further,
it indicates the birth of chaotic oscillations for larger values of through an àla
period-doubling cascade.

Figure 5. Bifurcation diagram forgenerated by equation (30)Berezowski transformation of the discrete-time equation for the `-monopolist
can be formulated from (17). First, we rewrite it asx(t) = x(t1 ) + K fa

where, for= 3; g[x(t1 ); x(tx3 (t21 ) + x (t1 )x(tcbg[x(t1 ); x(t

2 )]g

(46)

2 )] has the form of21 )x (t2 ) + x(t32 ) + x (t2 ):Replacing the left-hand side of (46) by x(t)_ + x(t); we have a continuous-time
output adjustment equation in Berezowski’s sense,
x(t)
_
=

x(t) + x(t1 ) + K facbg[x(t1 ); x(t2 )]g :

(47)

This is a nonlinear delay di¤erential equation, which is reduced to the di¤erence
equation (46) if = 0. Its linearized version is
x(t)
_
=

x(t) + 1

A
2

x(t

1)

A
x(t
2

and its characteristic equation for x(t) = e t u is
2

+2

(2

A)e
18

1

+ Ae

2

= 0:

2)

(48)

Dividing each term by 2(1 +a1 ( ) =Hence
a1 (i!) =anda2 (i!) =); we get again equation (32) withA 2
A
and a2 ( ) =2(1 + )
2(1 +

A 2
A 2
=
2
2(1 + i! )2 1 + ( !)

i

A
A
=
2
2(1 + i! )2 1 + ( !)

i

)

:

! (A

2)
2

2 1 + ( !)!A

2

2 1 + ( !)implying that2ja1 (i!)j =and(A 2)2hi
2
4 1 + ( !)2ja2 (i!)j =

h

A2
2

4 1 + ( !)i:

The triangle conditions (33) and (34) have the forms
jA 2j
A
q
+ q
2
2
2 1 + ( !)2 1 + ( !)and1jA 2j Aq
2
2 1 + ( !)

1

1

(49)

(50)

Now we have to consider two cases:(i) 0 < A

2

In this case, the left-hand side of (49) gives1
ja1 (i!)j + ja2 (i!)j = q< 1:
2
1 + ( !)The last inequality violates the direction of the inequality of (49), so there is no
stability switch.
Theorem 7 If 0 A
with all 1 0 and 2

2, then the steady state is locally asymptotically stable0:(ii) A > 2

19

This condition is equivalent to A
gives

2 > 0; then the left-hand side of (49)A

ja1 (i!)j + ja2 (i!)j = q

1

12

1 + ( !)

or

A(A

!2

2)
2

showing that this condition holds if
0<!

!1 =

Similarly, condition (50) has the form
1

which always holds.

1p
A(A

1
q
2
1 + ( !)

2):

(51)

1

Theorem 8 If A > 2, then stability switch might occur with all ! values satisfying relation (51).
Similarly to the previously discussed model, based on Figure 3, the application of the law of cosine presents,
102(A 1) A
1 @ 1 + ( !)q(52)
1 (!) = cos
2jA 2j 1 + ( !)and2 (!) = cos

02

1 @ 1 + ( !)

jAj

q

+ (A
2

1 + ( !)

1

1) A

:

(53)

Notice that 1 2 [0; ] and 2 2 [0; =2] ; furthermore, the arguments of a1 (i!)
and a2 (i!) arearg [a1 (i!)] = tan 1 ( !) + 2
andarg [a2 (i!)] =tan 1 ( !) + 2 :At points (0; 0) and (1; 0) in Figure 3, the angle balance relations imply thefollowings,
1(arg [a1 (i!)] + (2m 1)(54)
1 (!))
1;m (!) =!
and1(arg [a2 (i!)] + (2n 1)(55)
2 (!))
2;n (!) =!
20

where m and n are integers such that both 1 and 2 are nonnegative. Similarly
to the previous case, there are in…nitely many stability switching curves. Some
curves are illustrated in Figure 6(A), where the locus of ( 1;m (!); 2;n (!)) in
the feasible region are shown for ! 2 [0; ! 1 ] and m; n = 0; 1. The loci of( 1;0 (!); +2;0 (!)) is the bold black curve that divides the yellow region into
the stability and instability regions. Two numerical results are depicted in0
Figure 6(B). Taking 2 = 1, we simulate equation (47) for 1 2 [ a1 ; a1 ] with0aa
It is seen that1 ' 0:183 and 1 = 1 to obtain the blue bifurcation diagram.
0
a multiple-periodic cycle is possible for 1 closer to a1 . The red bifurcation
diagram reproduces a part of the bifurcation diagram of Figure 6 for 1 2
00[ b1 ; b1 ] with b1 ' 0:295 and b1 = 1:2.

(A) Feasible region(B) Bifurcation diagram

Figure 6. Dynamics generated by (42) amd (46)6Concluding Remarks

In this paper gradient adjustment process was introduced in a limited monopoly,when it does not know an analytic form of its pro…t function but can observe theactual pro…t at any time. The marginal pro…t was approximated with a simple…nite di¤erence formula based on two past pro…t observations leading to dynamic
models with two time delays. For the sake of comparison, a knowledgeable
monopoly knowing the form is also introduced. In the discrete-time adjustment
process, two main results are obtained: (1) The stability condition is the samefor both monopolists; (2) After losing stability, the knowledgeable monopolist
experiences various dynamics ranging from periodic oscillations to chaos as theadjustment coe¢ cient increases, whereas the limited monopolist produces only
periodic cycles.

21

Two di¤erent delay continuous dynamics were then constructed based on thediscrete-time model and analyzed to derive the stability switching curves. In
the …rst continuous-time model with Euler approximation, two main results are
summarized as follows: (3) The stability threshold value of 1 for the limitedmonopolist is less than that of the knowledgeable monopolist; (4) After losing
stability, both monopolists present only periodic oscillations. In the second
continuous-time model with Beresowski transformation, two main results are
the followings: (5) The stability threshold value of 1 for the limited monopolist
is less than that of the knowledgeable monopolist due to the negative sloping of
the stability switching curve. This is the the same as result (3) above; (6) Afterlosing stability, the limited monopolist can produce at most a multi-periodic
cycle, where as the knowledgeable monopolist goes into chaotic oscillations,passing through an àla period doubling cascade.
In approximating the marginal pro…t a very simple di¤erentiation formula
was used. However, more sophisticated formulas could provide better approximations with increased numbers of delays. It is an interesting task to see how
the more sophisticated di¤erentiation formulas alter the stability conditions.

22

References[1] Askar, S., On complex dynamics of monopoly market, Economic Modelling,31, 586-589, 2013.
[2] Baumol, W. and Quandt, R., Rules of thumb and optimally imperfectdecisions, American Economic Review, 54, 23-46, 1964.
[3] Berezowski, M., E¤ect of delay time on the generation of chaos in continuous systems. One-dimensional model. Two-dimensional model-tubularchemical reactor with recycle, Chaos, Solitons and Fractals, 12, 83-89, 2001.
µ[4] Cermák,
J. and Jánský, J., Explicit stability conditions for a linear trinomial delay di¤erence equation, Applied Mathematics Letters, 43, 56-60,
2015.
[5] Clower, R., Some theory of an ignorant monopolist, Economic Journal, 69,
705-716, 1959.
[6] Elsadany, A. and Awad, A., Dynamical analysis of a delayed monopoly
game with a log-concave demand function, Operations Research Letters,
2015, http://dx.doi.or/10.101y/j.orl.2015.11.001.
[7] Farebrother, R., Simpli…ed Samuelson conditions for cubic and quartic
equations, The Manchester School of Economics and Social Studies, 41,398-400, 1973.
[8] Gu, K., Niculescu, S-I. and Chen, J., On stability crossing curves for generalsystems with two delays, Journal of Mathematical Analysis and Applications, 311, 231-253, 2005.
[9] Matsumoto, A. and Szidarovszky, F., Nonlinear delay monopoly withbounded rationality, Chaos, Solitons and Fractals, 45, 507-519, 2012.
[10] Matsumoto, A. and Szidarovszky, F., Complex dynamics of monopolies
with gradient adjustment, Economic Modelling, 42, 220-229, 2014.
[11] Naimzada, A. and Ricchiuti, G., Complex dynamics in a monopoly with arule of thumb, Applied Mathematics and Computation, 203, 921-925, 2008.
[12] Puu, T. The chaotic monopolist, Chaos, Solitons and Fractals, 5, 35-44,
1995.

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