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

Discussion Paper No.210

Discrete and Continuous Dynamics in Nonlinear
Monopolies
Akio Matsumoto

Ferenc Szidarovszky

Chuo University

University of Pécs

September 2013I

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Discrete and Continuous Dynamics in Nonlinear
Monopolies
Ferenc Szidarovszkyz
University of Pécs

Akio Matsumotoy
Chuo University

Abstract
Dynamic monopolies are investigated with discrete and continuous
time scales by assuming general forms of the price and cost functions.
The existence of the unique pro…t maximizing output level is proved. The
discrete model is then constructed with gradient adjustment. It is shownthat the steady state is locally asymptotically stable if the speed of adjustment is small enough and it goes to chaos through period-doublingcascade as the speed becomes larger. The non-negativity condition thatprevents time trajectories from being negative is derived. The discretemodel is converted into the continuous model augmented with time delay and inertia. It is then demonstrated that stability can be switched
to instability and complex dynamics emerges as the length of the delay
increases and that instability can be switched to stability as the inertiacoe¢ cient becomes larger. Therefore the delay has the destabilizing e¤ect
while the inertia has the stabilizing e¤ect.

Keywords: Time delay, Inertia, Gradient dynamics, Continuous anddiscrete dynamics, Bounded rationality, Monopoly

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

The literature on monopolies and oligopolies plays a central role in mathematical
economics. The existence and uniqueness of the equilibrium were the researchissues in early stage and then dynamic extensions become the main topic of researchers. Linear models were …rst examined, where local asymptotical stability
implies global stability. Each model is based on a particular output adjustment
scheme. In applying best response dynamics, global information is needed about
the pro…t function while in the case of gradient adjustments only local information is needed to assess the marginal pro…t. The early results on static anddynamic oligopolies are summarized in Okuguchi (1976) and their multiproductgeneralization are discussed in Okuguchi and Szidarovszky (1999). During thelast two decades an increasing attention has been given to nonlinear dynamics.
Bischi et al. (2010) gives a comprehensive summary of the newer development.
As a tractable case many authors have examined dynamic monopolies. Baumol and Quandt (1964) have investigated cost-free monopolies, the dynamic
extensions of which were examined in both discrete and continuous time scales.
Their adjustment scheme resulted in convergent processes. Puu (1995) assumed
cubic price and linear cost functions and considered only discrete time scales.
Naimzada and Ricchiuti (2008) assumed cubic price and linear cost functions,and their model was generalized by Askar (2013) with a general from of theprice function keeping the linearity of the cost function. It is shown in thesestudies that chaotic dynamics arises via period doubling bifurcation.
In this paper, we reconsider monopoly dynamics in three di¤erent points ofview. First, we generalize the model of Askar (2013) by introducing more general
types of the cost function and study local and global stability with the nonnegativity condition that prevents time-trajectories from being negative. The
discrete-time model is converted into the continuous-time model by introducing
a time delay and an inertia in the direction of output change. Second, after
examining the stability with continuous time scale, we reveal that the discretemodel is less stable than the continuous model. Lastly, we numerically andanalytically show that the continuous model gives rise to complex dynamics
involving chaos due to delay and inertia.
The paper is organized as follows. In Section 2, the discrete-time model ispresented, conditions are derived for local asymptotic stability. In Section 3,
the continuous-time model is constructed from the discrete-time model. Stability with respect to time delay and inertia is considered. In the …nal section,
concluding remarks are given.

2

Discrete Time ModelIn this section we construct a discrete time dynamic model of a monopoly,
after determining the pro…t maximizing output. Consider a monopoly, where
q is its output, p(q) = a bq the price function (a; b > 0;1) andC(q) = cq (c > 0;2) its cost function. The pro…t of the monopoly is given2as(q) = (a

bq )q

cq :

b( + 1)qc q

By di¤erentiation0(q) = a

1

(1)

and
00

(q) =

b ( + 1)q

1

c (

1)q

2

< 0:

So (q) is strictly concave in q, and since 0 (0) = a > 0 and limq!1 0 (q) = 1;
there is a unique positive pro…t maximizing output, q; which is the solution of
the …rst-order conditionb( + 1)q + c q

1

= a:

(2)

The left hand side is strictly increasing, its value is 0 at q = 0 and converges
to in…nity as q ! 1, therefore the unique positive solution can be obtained bysimple computer methods (see, for example, Szidarovszky and Yakowitz 1978).
In the numerical considerations to be done below, we always use the following
speci…cation of the parameters,
a = 4; b = 3=5 and c = 1=2
which are also used by Naimzada and Ricchiuti (2008) and Askar (2013). Equation (2) is depicted by the roughly-meshed surface shown in Figure 1 where theblack dots on the surface represent the values of q for integer values of andin interval [2; 8]. It can be seen that the pro…t maximizing output is decreasing
in and :Figure 1. Determination of the
optimal q3

In applying gradient dynamics it is assumed that the monopoly adjusts its
output level in proportion to the marginal pro…t,q(t) = k 0 (q(t))

q(t + 1)

where k > 0 is the speed of adjustment. Substituting equation (1) results in thefollowing adjustment equation of output:
q(t + 1) = q(t) + k ab( + 1)q(t)1c q(t):(3)If= 1 and= 2; then this equation is linear with uninteresting global
dynamics. So will assume that either> 1 or> 2 or both occur. So
under this assumption, equation (3) is a nonlinear di¤erence equation, the local
asymptotic stability of which can be examined by linearization. Equations (2)and (3) imply that the steady state of this system is the pro…t maximizing
output q. Linearization of the right hand side around q results in the linear
equationq (t + 1) = 1
with q (t) = q(t)

+c (

1)q

2

and B = c (1)q2

k b ( + 1)q

1

q (t)

q. Let1

A = b ( + 1)q;

then we haveq (t + 1) = (1k(A + B)) q (t):The steady state of (3) is locally asymptotically stable if
j1

k(A + B)j < 1

j1

k(A + B)j > 1:and locally unstable if
Thus we have the following result on local stability:
Theorem 1 The steady state of system (3) is locally asymptotically stable if
k < k S and locally unstable if k > k S where
kS =

2
:
A+B

If we write di¤erence equation (3) in function iteration form, then it becomesq(t + 1) = '(q(t))where'(q) = q + k ab( + 1)q

4

c q

1

with

and

8
1
>
>
>
>
<
1
'0 (0) =
>
>
>
>
:
1

if

> 1 and

> 2;

1)

if

> 1 and= 2;kb( + 1)if

= 1 and>2

kc (lim '0 (q) =q!1

1 and '00 (q) < 0:

Since the right hand side of di¤erence equation (3) takes a unimodal pro…le,the steady state could proceed to chaotic ‡uctuations via the period-doublingbifurcation if '(q) has strong nonlinearities and its steady state is locally unstable. Naimzada and Ricchiuti (2008) numerically con…rm the generation ofcomplex dynamics under = 3 and = 1. Later Askar (2013) extends theiranalysis to the case with
3: We also conduct numerical simulations on
global behavior in the more general case of
1 and
2: Figure 2 is abifurcation diagram with= 4 and= 2 in which the steady state loses
stability at k = k S (' 0:154): It is seen that the trajectories exhibit periodic
and aperiodic ‡uctuations and are economically feasible (i.e., non-negative) fork < k N (' 0:228): In Appendix we consider the non-negativity condition thatguarantees the non-negativity of the trajectories for all t 0 and determine thevalue of k N : The main result obtained there is summarized as follows:
Theorem 2 There is a positive k N such that R(qm (k N ); k N ) = qM (k N ) andthe non-negativity condition
R(qm (k); k)qM (k)

holds for k
k N where, given k; R(q; k) = '(q); qm (k) maximizes '(q) forq 0 and qM (k) solves '(q) = 0:5

Figure 2. Bifurcation diagram of (3) withrespect to k

3

Continuous Time ModelIn this section we consider continuous-time dynamics based on the discrete-timemodel. Notice …rst that the discrete system can be rewritten as[q(t + 1)q(t)] + q(t) = '(q(t)):

There are many ways to transform a discrete-time model into a continuous-timemodel. One of the simplest ways is to replace the di¤erence, q(t + 1) q(t); withthe derivative q(t);_q(t)_ = q(t) + '(q(t))orq(t)_ = k 0 (q(t)):
Since 00 (q) < 0; this continuous-time model is always locally asymptoticallystable. Furthermore, its trajectory cannot be negative.1 Another simple way inobtaining a corresponding continuous model is the following. We …rst assume
delayed adjustment (i.e., '(q(t))) and second, an inertia in the direction of
output change (i.e., q(t + 1) q(t) = q(t))_
to get the delay di¤erential equation
q(t)
_ =

q(t) + '(q(t1 When))it goes to negative from positive, the trajectory has to go through zero whereq(t)_= ka > 0: So the trajectory turns back to the positive region.

6

or
q(t)
_ =

q(t) + q(t)+k ab( + 1)q(t

)

c q(t

)

1

(4)

where > 0 is a constant and > 0 is the delay.2 It is clearly seen that the delaydi¤erential equation (4) reduces to the di¤erence equation (3) in discrete time
step if = 0. This suggests that the evolution in discrete time might describe
well the evolution in continuous time if is small enough. In the following, we
pursue a possibility that the delay contiuous-time model gives rise to complex
dynamics when its discrete version also generates complex dynamics and shedlight on the condition under which this is a case.
The linearization with respect to q(t) results in the linear equation
q_ (t) =q (t) + (1k(A + B)) q (t):

In order to get the characteristic equation, assume that q(t) = e t u; then+1(1k(A + B)) e

= 0:

(5)

We …rst show that the roots of the characteristic equation (5) are single. On
the contrary, suppose that is not single. Then is a root of its derivative+ (k(A + B)

1)e

(

)=0

(k(A + B)

1)e

=

:

from which
Substituting this relation into equation (5) gives+1+=0

where the only multiple eigenvalue can be
=

1

1

:

Consequently all pure complex roots are single.
If = 0; thenk
(A + B) < 0:=Hence the steady state is locally asymptotically stable if there is no delay (i.e.,= 0) regardless of the value of the speed of adjustment. If = 0; then thecontinuous system is reduced to the discrete system and its stability depends
on the value of the speed of adjustment as con…rmed in Theorem 1. Our mainconcern in this section is placed on two issues: one is whether a positive delay
destabilizes the stable steady state and the other is whether a positive inertia
2 For this transformation, see Berezowski (2001) where the logistic map is connected withsome physical process of de…nite inertia.

7

stabilizes the unstable steady state. The remaining part of this section is divided
into two parts, the …rst issue is considered in the …rst part and so is the second
issue in the second part. Before proceeding, Figure 3 depicts a bifurcation
diagram of the delay di¤erential equation (4) with respect to k similarly to
Figure 2. For each value of k the equation is simulated for 2000 times where= 2 and= 0:1. The data for 1900 < t < 2000 are plotted verticallyabove the value of k: There are similarities and dissimilarities between this…gure and Figure 2. For relatively smaller values of k, the continuous system isstable and thus trajectories converge to the stationary state. For values greaterthan the threshold value of k1S ; a limit cycle appears with period-doubling andthen bifurcates to a period-8 limit cycle with two-times doubling at the second
threshold value of k2S . For even larger values of k, dynamic behavior of output
is very aperiodic. So we will look more carefully into how the delay a¤ects thisbifurcation process.

Figure 3. Bifurcation diagram of (4) withrespect to k

3.1

Delay E¤ect

In order to …nd stability switches in the case of > 0; we assume thati! with ! > 0. Substituting it into equation (5) yields
i! + 1(1k(A + B)) (cos !

=i sin ! ) = 0:

By separating the real and imaginary parts, we get two equations for the two
unknowns and !:1[1! + [1k(A + B)] cos ! = 0;k(A + B)] sin ! = 0;8which can be written as
cos ! =
sin ! =1;k(A + B)

1

(6)

!
:
k(A + B)

1

Adding the squares of the two equations and arranging the terms, we have[12

k(A + B)] = 1 + ( !)2

or
!2 =

[1

2

k(A + B)]

1

2

:

(7)

If j1 k(A + B)j < 1; then there is no positive solution of !, therefore there
is no stability switch, that is, the steady state is locally asymptotically stablewith arbitrary length of the delay. This is the case, whenk < kS =2:
A+B

Notice that this is the stability condition in the discrete model. This follows
the general understanding that a continuous-time model is more stable than adiscrete-time model in the sense that the former stability region in the parameter
space is larger than the latter’s.
Lemma 1 If the discrete-time model (3) is locally asymptotically stable, thenso is the continuous-time model (4).
Assume next that k > k S or 1 k(A + B) < 1. Then from (7) the solution
for ! isq2
[1 k(A + B)]
1
!=
:
(8)
Since 1 k(A + B) is negative, equations (6) imply that sin ! is positive andcos ! is negative, so =2 + 2n
! < + 2n and
n =

1
!

sin 1

!
k(A + B)1

+ 2n

for n = 0; 1; 2; :::(9)

We will next examine the direction of the stability switches. By selecting
as the bifurcation parameter, we verify how the change in the length of the delaya¤ects the real parts of the roots of the characteristic equations. Consideringas the function of ; = ( ); and implicitly di¤erentiating the characteristic
equation (6), we havedd(1k(A + B))e

9

d
d

= 0:

For the sake of analytical convenience, we solve for (d =d ) 1 ,
d
d

1

=

(

+ 1)

(10)

where the relation+ 1 = (1 k(A + B)) e
from (5) is used to obtain theform at the right hand side. Substituting = i! with ! > 0 into equation (10)and then taking the real part represent#"
1( ! + i)
d
= Re
Re
d
!( ! i)( ! + i)
2=( !)2 + 1

> 0:

The last inequality implies that increasing induces the crossing of the imaginary axis from left to right. So stability is lost at the critical delay with n = 0 in
equation (9),
!!
s
1
1
sin
1
: (11)
0 = q
2
2[k(A + B) 1][k(A + B) 1]
1

and stability cannot be regained later. In short, we summarize the results asfollows:

Lemma 2 In the case of k > k S ; the continuos model (4) is locally asymptotically stable if < 0 :Lemmas 1 and 2 together imply the following:
Theorem 3 Given k > 0; the delay di¤ erential equation (4) is locally asymptotically stable and delay is harmless if eitherk < kSork > k S and<0:

We now numerically examine global dynamics when the stability of thesteady state is lost. Figure 4(A) depicts the downward-sloping partition curve
described by equation (11) with= 3 and= 1:3 The curve divides the(k; ) plane into two regions. The steady state is locally unstable in the region
above the curve and locally asymptotically stable in the yellow region below the
curve. This stable region is further divided into two by the vertical dotted line
3 Needless to say, speci…ed values ofresults to be obtained below.

and

10

do not a¤ect the qualitative aspects of theat k = k S : In the region left to the line, k < k S holds and thus the steady stateis locally stable regardless of the length of delay. Such a delay is often called
harmless. In the region right where k > k S ; we select the value k0 (= 0:29)
and depict the vertical dotted-real line at k = k0 crossing the partition curve at= 0 : Lemma 2 implies that the steady state is locally asymptotically stablefor < 0 : On the partition curve, the real part of one eigenvalue is zero and its
derivative with respect to is positive. If becomes larger than 0 by crossing
the dotted vertical line at 0 from left to right, then one pair of complex eigenvalues changes the sign of their real part from negative to positive. Thus increasing> 0 destabilizes the model and the steady state bifurcates to a limit cycle asshown in Figure 4(B). At = 1 ; another pair of eigenvalues does the same. In
consequence, a new limit cycle emerges from the existing limit cycle with doubling the period of the cycle.4 So period-doubling bifurcation occurs. In addition
to this, carefully observing Figure 4(B) indicates the emergence of another new
limit cycle for 1 < < 2 .5 Thus the period of the limit cycle is more thandouble. At the next critical value, = 2 ; a new pair of eigenvalues changes realpart from negative to positive again and the existing cycle bifurcates to a newlimit cycle with doubling period and another new limit cycles are born. In thisway the bifurcation proceeds. If k 1 < < k ; then exactly k pairs have positive real parts and if ! 1; then k ! 1; so by increasing the values of ; thedynamics of the system becomes more and more complex generating complex
dynamics involving chaos. This is well illustrated in the bifurcation diagram in
Figure 4(B) in which we call such bifurcation quasi period-doubling because theperiod is increased to more than double. Global behavior with respect to issummarized as follows:
Proposition 1 Given k; the steady state loses stability at =to chaos via the quasi period-doubling cascade as increases.

0 and bifurcates

4 The values ofi for i = 1; 2; :::; 5 are determined by a rule of thumb.
5 The same phenomeon is often observed in a delay di¤erential equation. However, it is notclear yet why the new cycles arise suddenly.

11

(A) Partition curve(B) Bifurcation diagram

Figure 4. Delay e¤ect

3.2

Inertia E¤ect

In the same way as in the previous section, we can detect the e¤ect caused by achange in on dynamics. We …rst deal with the benchmark case of = 0: The
characteristic equation is
1 + (k(A + B)
If=

+ i with

1) e

= 0:

> 0; then it is written as

1 + (k(A + B)

1) e

(cos

Notice that stability depends on the sign ofparts yields two equations
1 + (k(A + B)

1) e

i sin

) = 0:: Separating real and imaginary
cos

=0

(12)

and
(k(A + B)
sin
cos

1) e

sin

= 0:

(13)

= 0 is de…nitely determined from (13). On the other hand, the sign ofin (12) is not determined unless the sign of k(A + B) 1 is speci…ed,
if k(A + B)

1 < 0, then cos> 0 implying cos=1

if k(A + B)

1 > 0, then cos< 0 implying cos

=

and

12

1:

So from (12),
1 + (k(A + B)1) e(+1) = 0 if k(A + B)

1 < 0;

1 + (k(A + B)1) e( 1) = 0

1 > 0:

if k(A + B)

Then
e

=

1
k(A + B)1

according to sign[k(A + B)

Since double-sign is in the same order,1] => 0 if and only if either(i) k(A + B)

1 < 0 and k(A + B)1<1(ii) k(A + B)

1 > 0 and k(A + B)

1 > 1:

orCondition (i) cannot occur as A > 0 and B > 0. It is therefore shown that thereal part is negative if k < k S and positive if k > k S : Hence, the stability
condition in the continuous model with = 0 is the same as the one in thediscrete model:
Theorem 4 The dynamic equation (4) withstable if k < k S and locally unstable if k:k S :of= 0 is locally asymptotically

Now we proceed to the case of > 0: If is given, then the critical values
can be obtained from equations (8) and (9),q2[k(A + B) 1]
1
:
(14)
q
n =
1
1
(2n + 1)sin
1 [k(A+B)
2
1]

They are the partition curves in the ( ; ) space. Largest n occurs at n = 0:n decreases in n and converges to zero as n ! 1: To verify whether stability
switch takes place on the partition curve, we suppose that = ( ): Implicitlydi¤erentiating the characteristic equation yields
d
+
d

(1

d
d

k(A + B))e=0

implying that
d
d
If

1

(

=

= i! with ! > 0; then the real part is
"
#
1
d
Re
= Re
d
=
13

+ 1)

:

( ! 2 i!)!2
< 0:That is, when the value of crosses a critical value from left to right, then one
pair of complex eigenvalues changes sign of real part from positive to negative.
Let 0 ( ) and 0 ( ) denote the right hand sides of equations (9) and (14) withn = 0: Both functions are strictly increasing. Assume that the value of is…xed and assume that > 0 = 0 ( ): Then = 0 ( 0 ) < 0 ( ); so Lemma 2
implies the local asymptotic stability of the continuous model with parametersand : When the value of crosses 0 from right to left, then one pair ofeigenvalues changes real part from negative to positive. Then at 1 ; anotherpair does the same, at 2 a new pair changes the sign of their real part fromnegative to positive, and so on. At each n the number of eigenvalues withpositive real parts increases by two when the value of crosses n from rightto left. So if n < < n 1 ; then exactly n pairs of eigenvalues have positive
real parts, that is, at any > 0 there are only …nitely many such eigenvalues
and as decreases, their number increases. Furthermore this number tends to
1 as ! 0: This is the reason of the "chaotic" behavior for small values asshown in the bifurcation diagrams of Figure 5(B). As increases, the number of
eigenvalues with positive real parts decreases making the dynamics of the model
more simple. This is also illustrated in Figure 5(B).
Proposition 2 Given ; dynamics becomes simple from complex via quasiperiod halving cascade as increases to 0 where stability is gained and neverlost for > 0 :(A) Partition curve(B) Bifurcation diagram

Figure 5. Inertia e¤ect
In summary we have the following result:
Theorem 5 The steady state of the continuous model (4) is locally asymptotically stable when the inertia does not a¤ ect stability if eitherk < kS14

ork > k S and

4

>

0:

Concluding Remarks

A discrete and corresponding continuous dynamics of a monopoly were examinedwith general price and cost functions. First the existence of the unique pro…tmaximizing output was proved and then stability conditions were derived forthe dynamic extensions. The discrete system is locally asymptotically stable
if the speed of adjustment is su¢ ciently small, in which case the continuous
system is also locally asymptotically stable with any length of the delay. It was
shown that the continuous system may become stable even in cases when thediscrete system is unstable with selecting su¢ ciently small length of the delay.
In examining global behavior we have seen a similarity between the dependence from the delay and the inertia coe¢ cient : The same quasi period
doubling phenomenon occurs with increasing values of and with decreasing
values of : This interesting fact was analytically proved and illustrated withnumerical studies.

15Appendix
In this Appendix, we consider the non-negativity condition for the trajectories. It is assumed that
1 and
2 where > 1 or > 2 or both. The
dynamic equation is given byq(t + 1) = R(q(t); k)
where
R(q; k) = q + k(ab( + 1)q

c q

1

)

(A-1)

with
R(0; k) = ka and lim R(q; k) =q!1

1:

Furthermore
R0 (q) = 1

k(b( + 1) q

1

+c (

1)q

2

):

(A-2)

Notice that R(q) is strictly concave.
Result 1 Given k > 0; there is a unique qM (k) such that R(qM (k)) = 0:
Proof. Equation (A-1) implies that qM is the solution of the equationb( + 1)q + c q

1

=a+

1
q
k

(A-3)

where the left hand side is denoted by f (q): At q = 0; the left hand side ofequation (A-3) is f (0) = 0 and is less than the right hand side. As q goesto in…nity, f (q) converges to 1 faster than the right 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. Furthermore R(q; k) > 0 if q 2 (0; qM ) andR(q; k) < 0 if q > qM :It is veri…ed from (A-3) thatlim qM (k) = 1, lim qM (k) = q

k!0

k!1

and
0
qM
(k) =

f 0 (q

(A-4)

q=k 2
>0
1=k
M)

where the inequality is due to the fact that at q = qM ; the left hand side crosses
the right hand side from below. In the same way we have the following.
Result 2 Given k; there is a unique qm (k) that maximizes R(q; k) for q

160:Proof. At any interior maximum,b( + 1) q

1

+c (

1)q

2

=

1
:
k

(A-5)

Let g(q) denote the left hand side, which strictly increase in q and tends to 1
as q ! 1. Notice that8b( + 1)
if = 1 and > 2;
>
>
>
>
<
c (
1) if > 1 and = 2;g(0) =>>>>:
0otherwise.

If g(0) < 1=k, then there is a unique solution qm (k) of equation (A-5). Furthermore, g(q) < 1=k as q < qm (k) and g(q) > 1=k as q > qm (k), so qm (k) is
the unique maximizer. If g(0)1=k, then g(q) > 1=k for all q > 0 implyingthat @R(q; k)=@q < 0; therefore qm (k) = 0 is the unique maximizer.
Notice that qm (k) satis…es the following limit relations:lim qm (k) = 1.

k!0

This is clear if g(0) < 1=k from equation (A-5), which becomes the case if k issu¢ ciently small. Furthermore
lim qm (k) = 0

k!1

which is also a simple consequence of equation (A-5) for interior maximum,
otherwise qm (k) = 0 for large enough values of k. Notice also that qm (k) strictly
decreases in k if it is positive and therefore solution of equation (A-5).
Result 3 qM (k) > qm (k) for any k > 0.
Proof. Since R(0; k) > 0 and R(q; k) 0 as q qM ; the maximizer qm has tobe less than qM ; so we have qM (k) > qm (k) for all k > 0:
If g(0) > 0; then there is a k1 such that g(0) = 1=k1 and if k < k1 ; theng(0) < 1=k, and if k
k1 ; then g(0) 1=k. So for k
k1 ; qm (k) = 0 and if
k < k1 ; then qm (k) is interior. If g(0) = 0; then qm (k) is interior for all k > 0;so we may select k1 = 1 in this case.
Result 4 R(qm (k); k) takes a U -shaped pro…le with respect to k and the qm (k)curve passes through its minimum point.
Proof. Assume k < k1 ; then qm (k) is interior. Di¤erentiating R(qm (k); k) withrespect to k gives
@R
dqm
@R
dR
=
+
dk
@q q=qm dk@k q=qm17

where the …rst term on the right hand side is zero at q = qm : So
dR
dk

= a

b( + 1) (qm (k))

1
(R(qm (k))
k=c (qm (k))

1

qm (k))

where equation (A-1) evaluated at q = qm (k) is used at the second step. The
pro…t maximizing output q is determined by f (q) = a and independent fromk: Since qm (k) is strictly decreasing in k while positive, limk!0 qm (k) = 1 andlimk!1 qm (k) = 0; there is a unique threshold value k such that qm (k) = q: In
consequence, since qm (k) > q for k < k, we have R(qm (k); k) < qm (k) thatthen leads to dR=dk < 0: In the same way, since qm (k) < q for k > k, we haveR(qm (k); k) > qm (k) that then leads to dR=dk > 0: Therefore the R(qm (k); k)
curve takes the U -shaped pro…le for k < k1 and the qm (k) curve passes theminimum value.
From Results 1-4, we have the following condition for the non-negativity ofthe trajectory which is given earlier in Theorem 2.
Result 5 There is a positive k N such that R(qm (k N ); k N ) = qM (k N ) and thenon-negativity condition
R(qm (k); k)
holds for kqM (k)

kN :Proof. For k < k;R(qm (k); k) < qm (k) < qM (k):For k > k; R(qm (k); k) is increasing in k and converges to 1 as k ! 1 whileqM (k) is decreasing. Thus there is a unique k N > k where R(qm (k N ); k N ) =qM (k N ) such thatR(qm (k); k)

qm (k) < qM (k) for kR(qm (k); k) < qM (k) for k < k < k N
andR(qm (k); k) > qM (k) for k > k N :18

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