IERCU
Institute of Economic Research, Chuo University
50th Anniversary Special Issues

Discussion Paper No.209

Complex Dynamics of Monopolies with
Gradient Adjustment
Akio Matsumoto

Ferenc Szidarovszky

Chuo University

University of Pécs

September 2013I

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Complex Dynamics of Monopolies with
Gradient Adjustment
Ferenc Szidarovszkyz
University of Pécs

Akio Matsumotoy
Chuo University

Abstract
This paper aims to show that delay matters in continuous- and discretetime framework. It constructs a simple dynamic model of a boundedly
rational monopoly. First the existence of the unique equilibrium stateis proved under general price and cost function forms. Conditions arederived for its local asymptotical stability with both continuous and discrete time scales. The global dynamic behavior of the systems is thennumerically examined, demonstrating that the continuous system is globally asymptotically stable without delay and in the presense of delay if thedelay is su¢ ciently small. Then stability of the continuous system is lost
via Hopf bifurcation. In the discrete case without delay, the steady stateis locally asymptotically stable if the speed of adjustment is small enough,then stability is lost via period–doubling bifurcation. If the delay is one
or two steps, then stability loss occurs via Neimark-Sacker bifurcation.

Keywords: Gradient dynamics, Boundedly rational monopoly, Continuous and discrete dynamics, Time delay

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 Scienti…c 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, Japan. 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

It is well known that economic agents are not fully rational, and usually prefer
to employ simple rules which were previously tested (Kahneman et al., 1986).
Many di¤erent output adjustment schemes have been developed. Bischi et al.
(2010) o¤er a collection of the most important schemes in the case of oligopolies.
In earlier studies linear models were examined, where local asymptotical stability
implies global asymptotical stability (Okuguchi and Szidarovszky, 1999). In
the last two decades an increasing attention has been given to examine theasymptotical property of nonlinear economic systems including monopolies andduopolies (Bischi et al., 2010). It is also well known that economic dynamic
systems inherently incorporate delays in their actions and delay is one of theessentials for economic dynamics. Nevertheless, little attention has been given
to studies on delay dynamics of boundedly rational economic agents. The main
purpose of this paper is to show how delay a¤ects dynamics in continuous- as
well as in discrete-time framework.
In this paper dynamics of boundedly rational monopolies are discussed withthe most popular adjustment rule in which the …rm adjusts output in proportion
to its marginal pro…t. Such an adjustment scheme is known as gradient adjustment. In contrary to best response dynamics, only local information is needed
for the adjustment process which is always available to the …rm. Baumol andQuandt (1964) investigated cost free monopolies in both discrete and continuous
time scales and the dependence of the pro…t on varying price. They developed
a simple adjustment mechanism that converges to the pro…t maximizing output. Puu (1995) has revisited this model with discrete time setting and a cubic
demand function. It is shown that complex dynamics can emerge if the pricefunction has a re‡ection point. Naimzada and Ricchiuti (2008) reconsidered
Puu’s model with linear cost and cubic price function and exhibited the birthof chaos through the period-doubling bifurcation even if the price function does
not have a re‡ection point. Their model was then generalized by Askar (2013)
with more general price functions. In both studies local asymptotical stability
was analytically examined and global dynamics by computer simulations.
We consider monopoly dynamics from three di¤erent points of view. First,
we are concerned with gradient dynamics in continuous-time framework whilemost of recent studies considered discrete-time dynamics. Second, we will further generalize the model of Asker (2013) by introducing a more general class of
cost functions and determine the non-negativity condition that prevents timetrajectories from being negative. Third, we devote a little more space to exhibit that "delay" discrete-time monopoly gives rise to complex dynamics via
Neimark-Sacker bifurcation while "non-delay" monopoly goes to chaos througha period-doubling cascade.
The paper is organized as follows. In Section 2, the model is presented
and the existence of the unique pro…t maximizing output level is proved. In
Section 3, the dynamic system with the gradient adjustment is constructedin continuous time scale. Without delayed information the continuous modelis always locally asymptotically stable, and the stability can be lost if only2

delayed revenue information is available. In Section 4, the continuous-timemodel is discretized. It is analytically shown and numerically con…rmed thatthe discrete-time model can give rise to aperiodic oscillations through a perioddoubling bifurcation or Neimark-Sacker bifurcation according to whether themodel involves "delay" or not. In the …nal section, concluding remarks are
given.

2

Continuous Model

A general inverse demand function as well as a general cost function are considered in a monopoly. Let p(q) = a bq be the price function and C(q) = cq be
the cost function. The case of = 3 and = 1 was examined by Naimzada and
Ricchiuti (2008), the more general case with any
3 and = 1 was discussed
by Askar (2013). We move one step forward from their studies and consider thecase where both and are greater than 1.
Assumption 1.

> 1 and> 2:

The pro…t of the monopoly is given as(q) = (a

bq )q

cq :

b( + 1)qc q(1)

Notice that0(q) = a

1

(2)

and
00

(q) =

b ( + 1)q

1

c (

1)q

2

< 0;

(3)

so (q) is strictly concave in q; furthermore(0) = 0; lim

q!1

(q) =

1 and

0

(0) = a > 0:

Therefore there is a unique pro…t maximizing output q which is the uniquesolution of equationb( + 1)q + c q 1 = a:(4)
The left hand side is zero at q = 0, converges to 1 as q tends to in…nityand is strictly increasing. So the value of q can be obtained by using simple
numerical methods (see, for example, Szidarovszky and Yakowitz, 1978). The
remaining of this section has two subsections. In Section 2.1, the condition of
local asymptotical stability is derived. In Section 2.2, we examine the e¤ects
caused by changing values of and on stability.

2.1

Stability

Assuming gradient dynamics, the …rm adjusts its output according to the following di¤erential equation:q(t)_ = k 0 (q(t))
3

or
q(t)
_ = k(ab( + 1)q(t)c q(t)

1):(5)

This is a nonlinear system with positive adjustment coe¢ cient k. The uniquesteady state of this system is the pro…t maximizing output q. Local asymptotic
stability can be examined by linearization. The linearized equation can be
written asq_" (t) = k 0 (q)q" (t)or1q_" (t) = k( b ( + 1)q

c (

2

1)q

)q" (t)(6)

where q" (t) = q(t) q. Since the multiplier of q" (t) in the right hand side isnegative, the steady state q is locally asymptotically stable. Let r(q) denote
the right hand side of equation (5). Notice that it strictly decreases in q andr(q) = 0: So r(q) < 0 if q > q and r(q) > 0 if q < q: Therefore if q(0) < q; thenq(t) strictly increases and converges to q, if q(0) > q; then q(t) strictly decreases
and converges to q, and if q(0) = q; then q(t) = q for all t 0: Hence system(5) is global asymptotically stable, and with q(0) > 0; the entire trajectory q(t)remains positive. Although this result is well known, we formally state it as it
is a benchmark of this study:
Theorem 1 For any k, the non-delay continuous time model (5) is locally andglobally asymptotically stable.
Assume next that the monopoly receives marginal revenue information witha positive delay > 0; which could be due to delay price information. Thensystem (5) becomes a delay di¤erential equation,q(t)_ = k(ab( + 1)q(t)

1c q(t)):The derivatives of the right hand side with respect to q(t@ q(t)
_
=
@q(t
)
and

@ q(t)_=@q(t)kb ( + 1)q(tkc (1)q(t))

2(7)) and q(t) are1;

so the linearized equation has the form
q_" (t) =kAq" (t)

kBq" (t)(8)

with
A = b ( + 1)q1

and B = c (

1)q

2

:

We know that the system is locally and globally asymptotically stable for =0: In order to …nd stability switches with increasing value of ; assume thatq" (t) = e t u; and substitute it into equation (8) to obtain:e t=kAe (t
4

)

kBe t

(9)

so the characteristic equation is a mixed polynomial-exponential equation+ kB + kAe
With any stability switch,

= 0:

(10)

= i! with ! > 0; so

i! + kA(cos !

i sin ! ) + kB = 0(11)

and separating the real and imaginary parts we get two equations for two unknowns ! and :kA cos ! = kB
(12)
kA sin ! = !:By adding the squares of these equations we have
! 2 = k 2 (A2B 2 ):(13)

Since there is no positive solution for ! if A B 0; no stability switch occurs
for any > 0: In such a case, delay is often called harmless. Positive solution
exists if A > B: From (12) we see that cos ! < 0 and sin ! > 0; therefore
from the second equation in (12),!
!
p
A2 B 21
1sin+ 2n(n 0):(14)n = p
A
k A2 B 2
In order to …nd the direction of the stability switches at these critical values, consider as a function of the bifurcation parameter ;= ( ); anddi¤erentiate the characteristic equation (10) with respect to to have
d
+ kAe
d

d
d

= 0:

(15)

Using equation (10) yields

Withd( + kB)=:d
1 + ( + kB)(16)(! 2 i!kB)(1 + kB
i! )
d
=
2
2
d
(1 + kB ) + (! )(17)

= i!; we haveand so

d
!2
=
> 0:
(18)
d
(1 + kB )2 + (! )2
The crossing of the imaginary axis is from left to right as increases. Consequently stability is lost at the smallest critical value!!
p
1
A2 B 2
1
sin
(19)
0 = p
A
k A2 B 2Re

and cannot be regained for any>

0 . Thus we have the following result.

5

Theorem 2 The steady state in the continuous system is locally asymptoticallystable if A
B or if A > B and 0< 0 whereas it is locally unstable if
A > B and > 0 : If = 0 ; then Hopf bifurcation occurs giving the possibilityof the birth of limit cycles.

2.2E¤ects ofandon Stability

We derive conditions to guarantee that A > B:(i) Assume …rst that
only if1>

2 (that is,b ( + 1)q +1

<

>c (

or
q > q1 :=+ 1); then A > B if andc (1)
b ( + 1)

1)

(20)

1
+1

(21)

which is the case if f (q1 ) < a where f (q) denotes the left hand side ofequation (4);(ii) If=+ 1; then A > B if and only ifb > c;(iii) And …nally, if

>

(22)

+ 1; then A > B if and only ifb ( + 1) > c (orq < q1 =11)qb ( + 1)
c (
1)

(23)

1
1

(24)

which occurs if f (q1 ) > a.
First, to see dynamic behavior generated by the delay di¤erential equation(7), we assume the same parameter speci…cations as in Naimzada and Ricchiuti(2008) and Askar (2013).
Assumption 2. a = 4; b = 3=5 and c = 1=2:
Second, to describe these conditions in the ( ; ) plane, we substitute q1 intof (q) to obtain the following form,
13g( ; ) = (1 + )
5

5 (
1)
6 ( + 1)

1+

1
+
2

5 (
1)
6 ( + 1)1+which is shown in Figure 1(A), where the red-coloured vertical plane is thesurface of ( ; ) satisfying = +1 and divides the 3D space into two subspace,6

> + 1 in one of them and < + 1 in the other. The graph of g( ; ) in
the former is colored in meshed yellow and the one in the latter is in meshedlight blue. The top surface of the 3D space is the cross-section at the height
being 4. Since b > c is assumed, q1 converges to zero when approaches + 1
from below and converges to 1 when approaches + 1 from above, that is,lim!( +1)c (1)
b ( + 1)

and

1
+1

=0
1

c (
1) +1
lim
= 1:
b ( + 1)!( +1)+The graphical expression of the divergence of q1 is given by the fact that themeshed yellow surface is asymptotic to the = + 1 plane. It then follows thatit crosses the top surface and its intersection is the locus described by g( ; ) = 4
or f (q1 ) = a: On the other hand, the graphical expression of the convergenceof q1 is that the meshed light blue surface crosses the bottom surface of the
3D space along the locus de…ned by g( ; ) = 0 or = + 1: It is apparent
from Figure 1(A) that g( ; ) < 4 or f (q1 ) < a when<
+ 1: Figure1(B) is a projection of Figure1(A) onto the ( ; ) plane in which the upper
positive-sloping curve describes f (q1 ) = a and the lower positive-sloping line
represents the locus of = + 1: Thus A > B in the colored regions of Figure
1(B), more precisely, condition (iii) holds in the yellow region that is de…ned byf( ; ) j f (q1 ) > a and > + 1g, condition (i) holds in the light-blue regionf( ; ) j f (q1 ) < a and < + 1g and condition (ii) holds on the positivesloping line = + 1: We mention the meanings of the …ve dotted points in
Figure 1(B) later.

(A). 3D graph of g( ; )(B). Division

Figure 1. The graphical representation of the conditions for A > B
In the case of A > B; equation (19) is the partition curve in the (k; ) spacethat divides the parametric space into two subregions, the stable region below
7

the curve in which the steady state is locally stable and the unstable regionin which it is locally unstable. We numerically con…rm how the values ofanda¤ect stability of the steady state. In the …rst example, we examine
the e¤ect caused by increasing
on stability. So
is increased from 3 to 5
by unity while is …xed at 2 as denoted by three dotted points horizontally
located at = 2 in Figure 1(B). The stable regions with = 3; 4; 5 are coloredin light blue, ocher and yellowish green, respectively and one is supersimposedon another in Figure 2(A). The boundary of each region corresponds to thepartition curve with corresponding value of : Comparison of these cases showthat the partition curve shifts downward as the value of increases. This implies
that increasing has a destabilizing e¤ect by shrinking the stable region. In thesecond example, is increased from 2 to 4 by unity while is …xed at 4 as shownby three dotted points vertically located at = 4 in Figure 1(B). The stableregions with = 2; 3; 4 are colored in ocher, green and purple, respectively andone is superimposed on another in Figure 2(B). The boundary of each region
corresponds to the partition curve with corresponding value of : We can seethat the partition curve shifts upward as the value of increases. This implies
that increasing value of has a stabilizing e¤ect enlarging the stable region.
We summarize these results as follows:
Proposition 1 Increasingstabilizes it.

destabilizes the steady state whereas increasing

A. E¤ect of increasing

B. E¤ect of increasing

Figure 2. The stabilizing e¤ects caused by parameter changes
Finally we numerically con…rm the birth of a limit cycle when the stability
conditions are violated by increasing the length of delay. Specifying= 3,= 2 and k = 0:2 and increasing the value of from ' 1:0361 to ' 1:386
1 For= 3 and= 2; the critical value

0 is approximated as 1:036478:

8with increment of 1=1000 , we simulate the dynamic equation (5) for 0 t
1000 and plot the local maximum and minimum of the generated trajectoryfor 950 t 1000 for each value of to obtain the bifurcation diagram withrespect to as shown in Figure 3(A). This diagram implies that a limit cycle
emerges for > 0 and its diameter (di¤erence between the local maximum
and minimum) becomes larger as increases. It also indicates that a muchlarger value of (approximately, larger than 1.39 in this example) generates a
trajectory having a negative minimum and thus makes dynamics economicallyuninteresting. Taking = 1 ' 1:236; we depict a pair of (q(t); q(t)) for
950 t 1000 in Figure 3(B) to show the birth of limit cycle. Notice that thedi¤erence between the upper intersection of the upper branch with the verticaldotted line at = 1 and the lower intersection of the lower branch in Figure3(A) corresponds to the di¤erence between the maximum and minimum shownin Figure 3(B). These numerical examples con…rm Theorem 2.

A. Bifurcation diagram

B. Phase diagram

Figure 3. Emergence of a limit cycle

3

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. To this end, we discretize the delaydi¤erential equation (7) by replacing q(t)_ with q(t + 1) q(t) to obtainq(t + 1) = q(t) + k(ab( + 1)q(t)

c q(t)

1)(25)

and then reconsider local and global dynamics in discrete time. The steadystate of this di¤erence equation is the same as q of the di¤erential equation. We
mention that this discrete-time equation has a -step delay when
1.2 The2 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.

9remaining of this section is divided into two parts. In the …rst part, we examine
the case of = 0 and give a more precise account of the non-negativity condition.
In the second part, we discuss the case of
1 in detail to concentrate on delaye¤ects in the discrete-time framework.

3.1
If

No-delay:=0= 0; then equation (25) becomes a nonlinear …rst-order di¤erence equationq(t + 1) = q(t) + k(ab( + 1)q(t)1c q(t))(26)

where the right hand side is denoted by R(q(t)): The linearized equation of (26)
has the formq" (t + 1) = 1

k(b ( + 1)q

1

+c (

1)q

2

) q" (t)(27)

orq" (t + 1) = [1k(A + B)] q" (t):(28)

The steady state is locally asymptotically stable if
j1

k(A + B)j < 1

(29)

2
:
A+B

(30)

or
k < k S :=As expected, the discrete-time equation (27) has a more restrictive stability condition than the continuous-time equation (6).3 Indeed, the former has a criticalvalue of the speed of adjustment and is locally asymptotically stable if the speedof adjustment is smaller than the critical value and locally unstable if larger. On
the other hand, the continuous-time equation is locally asymptotically stable forany value of the speed of adjustment.
We now proceed to numerical simulations on global behavior when the steadystate is locally unstable. We …rst determine the non-negativity condition thatguarantees the non-negativity of the trajectories for all t 0 when
2 and
= 1 and then numerically verify it in the case of
2 and = 2:4 Solving
equation (4) with = 1 yields the explicit form of q,q=a cb( + 1)

1

:

(31)

In order to have positive solution of the …rst order condition we have to assume
that a > c.
Lemma 1 Given k; there is a unique qM (k) such that R(qM (k)) = 0:
3 It is well-known that the choice of time scales matters.
4 See Matsumoto and Szidarovszky (2013) for the non-negativity condition in the generalcase with
1 and
2.

10

Proof. R(qM (k)) = 0 implies that qM is the solution of equationb( + 1)q = (a

c) +

1
q
k

(32)

where the left hand side is denoted by F (q). At q = 0; F (0) = 0 and is less than
the right hand side. As q goes to in…nity, F (q) converges to 1 faster than theright hand side. So there is at least one solution. Since F (q) is strictly convex
and the right hand side is linear, the solution is unique.
It can be veri…ed, due to equation (32), thatlim qM (k) = 1;

k!0

lim qM (k) = q

k!1

and

dqM (k)qM =k 2=<0dk
1=k F 0 (qM )where F 0 (qM ) > 1=k because the F (q) curve intercepts the right-hand side ofequation (32) from below at q = qM :In the same way, we have the following.
Lemma 2 Given k; there is a unique qm (k) such that R0 (qm (k)) = 0:
Proof. R0 (qm (k)) = 0 implies that qm (k) is the solution of1b( + 1)q

=

1
k

(33)

where the left hand side is zero at q = 0; increasing and converges to 1 as
q ! 1 so the existence of the unique solution is clear.
Solving (33) gives the explicit form of qm (k),qm (k) =

1
k
D

with
D = ( b( + 1))

1
1

1
1

> 0:

It is then clear thatlim qm (k) = 1

k!0

lim qm (k) = 0

k!1

and

dqm (k)
=
dk

1
D(

1)

k

1

< 0;

so qm (k) strictly decreases in k. By these properties of qm (k); the following
results are clear.
11Lemma 3 There is a positive k such that qm (k) = q andqm (k) ? q for k 7 k:
The next lemma describes the relation between qM (k) and qm (k).
Lemma 4 qM (k) > qm (k) for any k > 0:
Proof. Notice …rst that equation (33) can be rewritten asb( + 1)q =

1
q:
k

(34)

Let G(q) denote the left hand side. Then from (32) we have
F (q) = (ac) +q
at q = qMkandq
at q = qm :kUnder > 1; G(q) > F (q) always for q > 0 and since a > c, the right handside of the …rst equation is larger than that of the second equation. ThereforeqM (k) > qm (k) follows for any k > 0:
G(q) =The last lemma con…rms the shape of the R(qm (k)) curve and the relation
with the qm (k) curve. In order to clarify the dependency, we denote R(qm (k))
as R(qm (k); k) and then introduce a new function, R(k) = R(qm (k); k):Lemma 5 R(k) takes a U -shaped pro…le with respect to k andR(k) Q qm (k) according to k Q k:
Proof. Di¤erentiating R(k) with respect to k yieldsdR@R(qm (k); k)dqm@R(qm (k); k)
=
+
dk
@q
@k
q=qm dkq=qm
where the …rst term on the right hand side is zero at q = qm : So
dR
dk

=a

c

b( + 1)qm(35)= b( + 1) (qwhere the equality ais then obtained thatqm )c = b( + 1)q from (31) is used at the second step. ItdR
Q 0 according to q Q qm ;dk12and by Lemma 3,

q Q qm () k Q k:

So R(k) takes a U -shaped pro…le and has its minimum at k = k: Notice thatfrom (25) and (35)dR
1
= (R(qm (k)) qm (k)) ;dkksoR(k) Q qm (k) for k Q kwhich means that the qm (k) curve passes the minimum point of R(k):Since substituting qm (k) into R(q) represents
R(qm (k)) = qm (k) + k(acb( + 1)(qm (k)) )and equation (33) can be written as
qm (k) = kb( + 1)(qm (k)) ;we haveR(qm (k)) = k(ac) + (

1)b( + 1)k (qm (k))

where
1
k

1
k (qm (k)) =
D

1
1

:

We then havelim R(qm (k)) = lim k(qm (k)) = 1

k!0

k!0

and
lim R(qm (k)) = lim k(a

k!1

k!1

c) = 1:R(qm (k)) is asymptotic to the vertical line when k approaches zero and to thek(a c) curve when k goes to in…nity.
From Lemmas 1 4; we have the following result.
Theorem 3 There is a unique value k N such that R(qm (k N )) = qM (k N ) andthe non-negativity condition
R(qm (k))
holds for kqM (k)k:

Proof. qM (k) > qm (k) and qm (k) > R(qm (k)) for k < k imply that R(qm (k)) <qM (k): For kk; R(qm (k))
q and increasingly converges to 1 whileqM (k) decreasingly converge to q: Thus there is a unique value k N such thatR(qm (k N )) = qM (k N ) and R(qm (k)) qM (k) for k k N :Solving R(qm (k N )) = qM (k N ) for k N and using (25) and (34) we have
13

kN =
Equation (30) withqM + (1)qm(c)= 1 and equation (31) yieldkS =2b( + 1)q1

and from (4)b( + 1)q = acimplying that
kS =

2q
:
(a c)

Taking = 2 and solving R(qm (k)) = qM (k) withupper bounds of k,= 3; 4; 5; 6 yield the

k3N ' 0:298; k4N ' 0:228; k5N ' 0:188 and k6N ' 0:162
where k N is the critical value k N for : Figure 4(A) depicts a locus of k N inthe ( ; k) plane and any trajectory generated by equation (26) with k 2 (0; k N )is non-negative for any t 0. Figure 4(B) presents a bifurcation diagram withrespect to k when = 4: It can be seen that the steady state is destabilized
when k = k S (' 0:154) and may be chaotic via the period-doubling bifurcation
as k increases. It may not be possible to obtain an explicit functional form of
3 as in the case of = 2, however, it is possible
k N in terms of and k forto calculate a particular solution of qM (k) as well as R(qm (k)) after specifying avalue of and thus to derive a bifurcation diagram as in Figure 4(B). Changing
the value of does not a¤ect the qualitative aspects of the results obtained with
= 4: The results in the case of = 0 are summarized as follows:
Theorem 4 Under Assumptions 1 and 2 with = 4 and = 2; the monopoly
equilibrium with = 0 is locally asymptotically stable if k < k S ; loses stability
if k = k S and arrives at complex dynamics involving chaos through a perioddoubling bifurcation if k S < k k N .

14(A) k N locus(B) Bifurcation diagram

Figure 4. Dynamic behavior with

3.2

Delay:= 2 and= 3; 4; 5; 6

1

If = 1; then equation (25) has one-step delay and then becomes a nonlinear
second-order di¤erence equationq(t + 1) = q(t) + k(ab( + 1)q(t1)

1c q(t))(36)

which can be converted to an equivalent 2D system of …rst-order di¤erence
equations,x(t + 1) = q(t);(37)
q(t + 1) = q(t) + k(ab( + 1)x(t)c q(t)

1):

The linearized system isx(t + 1)q(t + 1)

=

0
kA

1

1
kB

x(t)
q(t)

and its characteristic equation is transformed into a quadratic equation,
2

+ (kB

1) + kA = 0:(38)

The su¢ cient and necessary condition that this quadratic equation has roots
inside the unit cycle are given by the following three conditions,
1 + (kB

1) + kA > 0;1

(kB1) + kA > 0;

1

kA > 0:15(39)

The …rst condition of (39) is always satis…ed and so is the remaining two conditions if and only if eitherAB

0 and k <orA

B < 0 and k < min

1
A

1
2
;
A B A

:

As shown in Figure 1, we see the conditions under which pair ( ; ) gives rise
to A B > 0. Henceforth in order to simplify the analysis, we con…ne ourattention to the case where A B > 0 and put the following assumption.
Assumption 3. A

B > 0:

Destabilization of the monopoly steady state occurs only by violating thethird condition from which the threshold value is de…ned as
k S=1 =

1:
A

The monopoly equilibrium changes stability through a pair of complex conjugateroots. In particular, as k becomes larger than k S=1 , the steady state bifurcatesto a periodic cycle, which is then replaced with a quasi-periodic cycle. Sucha stability change is called Neimark-Sacker (NS henceforth) bifurcation. The
results analytically obtained are the following:
Theorem 5 Under Assumptions 1,2 and 3 with = 4 and = 2, the monopoly
equilibrium with = 1 is locally asymptotically stable if k < k S=1 ; loses stability
if k = k S=1 and generates aperiodic oscillations through a NS bifurcation if
k S=1 < k < k N=1 :Taking = 4 and = 2 and selecting k as the bifurcation parameter, we
illustrate the bifurcation diagram in Figure 5(A) in which stability of the steadystate is changed to instability at k = k S=1 (= 1=12) and cyclic behavior emerges
for k > k S=1 : When k arrives at k N=1 (' 0:119); the non-negativity condition isviolated resulting in the birth of economically uninteresting behavior. In Figure
5(B), we take k = k1 (= 0:11) and depict a phase diagram, that is, a cyclic
behavior in the (q(t 1); q(t)) plane.

16

(A). Bifurcation diagram(B). Limit cycle

Figure 5. Dynamic behavior via Hopf bifurcation
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 (25) is now a third-order di¤erence equation,q(t + 1) = q(t) + k(ab( + 1)q(t2)

1c q(t)):(40)

This can be written as a 3D system of …rst-order di¤erence equationsx(t + 1) = y(t)y(t + 1) = q(t)(41)

q(t + 1) = q(t) + k ab( + 1)x(t)c q(t)

1;

where the steady state is (x; y; q) with x = y = q: Linear approximation ofequation (41) yields the approximated system having the form
0
1 0
10
1
x(t + 1)0
10x(t)@ y(t + 1) A = @ 0
A @ y(t) A
01(42)q(t + 1)
kA 0 1 kBq(t)and the corresponding characteristic equation is cubic
3

+ (kB

1) 2 + kA = 0:(43)

The steady state is locally asymptotically stable if all eigenvalues of equation(43) are less than unity in absolute value. Farebrother (1973) and Okuguchi
and Irie (1990) have proved that the most simpli…ed form of the su¢ cient and17necessary conditions for the cubic equation to have roots only inside the unit
cycle are1 + a1 + a2 + a3 > 0;1

a1 + a21

a2 + a1 a3

a3 > 0;(44)
a23 > 0;a2 < 3where
a1 = kB

1; a2 = 0 and a3 = kA:

It can be veri…ed that the …rst and fourth conditions are always satis…ed while
the second and third condition holds if
k < k2S :=22pand k < k3S :=2
A+B
A + A + 4A(AB):

If we denote the left-hand side of the third condition in (44) by g(k); it can be
written asg(k) = A(B A)k 2 Ak + 1:
This quadratic polynomial has two real roots, one is negative and the other, k3S ;is positive. Substituting k2S into g(k) yields
g(k2S ) =(A

B)(5A + B)
< 0:
(A + B)2

This inequality implies that k3S < k2S : Numerical simulations are illustrated
in Figure 6 in which the stability is lost at k3S ' 0:053; the non-negativity
condition is violated at k N=2 ' 0:053: The phase diagram is illustrated for
k1 = 0:0675: Figure 6 is essentially similar to Figure 5. Therefore rewriting k3S
as k S=2 describes the results that have been made as follows:
Theorem 6 Under Assumptions 1,2 and 3 with = 4 and = 2, the monopoly
equilibrium with = 2 is locally asymptotically stable if k < k S=2 ; loses stability
if k = k S=2 and generates aperiodic oscillations through a NS bifurcation if
k S=2 < k < k N=2 :18

(A). Bifurcation diagram(B). Limit cyleFigure 6. Dynamic behavior of system (41)4Conclusion

Static and dynamic monopolies were examined with general price and cost functions. The existence of a unique pro…t maximizing output was …rst proved which
can be computed by solving a single-variable monotonic equation. Continuous
time scales were then assumed and the local asymptotic stability of the resulting
dynamic system was proved regardless of model parameter values. If time delay
was assumed in obtaining revenue information, then the system is locally asymptotically stable with arbitrary length of the delay for A B; and if A > B; thenlocal asymptotic stability occurs if the delay is smaller than a given threshold.
Notice that this threshold value decreases with increasing value of the speedof adjustment. Under discrete time scales the system is locally asymptoticallystable if the speed of adjustment is below a certain threshold which is decreasing
if the values of A and/or B increases. If the length of the delay is one step,then similar result holds where the threshold decreases in A but is independentof the value of B when B < A: If the length of the delay is two time steps, then
the threshold depends on both A and B; increases as A decreases and/or B
increases. It should be worthwhile to point out that in the discrete-time model
the steady state loses stability via period-doubling bifurcation if = 0 and via
Neimark-Sacker bifurcation if
1:

19

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