# Triumph over Your Rivals in Dynamic Oligopoly

Authors: Weihong Huang
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

Challenging the conventional belief that sophistication in strategy is always better, it was found in W.
Huang (2002a) that a price-taker who adopts the Cobweb strategy yields higher profits than those who adopt
more sophisticated strategies. This study explores the possibility of improving further the relative profit
advantage that the price-taker has over its counterparts through incorporating the growth-rate adjustment
strategy. A linear heterogenous oligopoly model is used to illustrate the merits of such strategy in the case of
disequilibrium. It is shown in theory and supported with numerical simulations that the adoption of growth-rate
adjustment strategy together with price-taking strategy confers on the price-taker the stabilization power
in a dynamically unstable market in addition to better relative performance in terms of major performance
measures.


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## Body

## 1. Introduction

One of the lessons learned from basic microeconomics
is that for a profit maximizing, firm will always leverage on the market
information as well as the behavioral rule of its rivals when making its output
decision. In other words, a firm who ignores its market power and simply
behaves a price-taker appears to be economically irrational. A typical example
of this belief is demonstrated by the classic Cournot model in which each
output decision of profit-maximizing firm is based on a best response function
that is derived using the firm's forecast about its rivals' output level.
However, such beliefs were first challenged in [1, 2] where an oligopoly
that consists of a price-taker and many sophisticated firms, with identical
technology, was studied. A counter-intuitive phenomenon is revealed—no
matter what strategies the sophisticated firms may adopt, the price-taker
always triumphs over them in terms of relative profitability at any
intertemporal equilibrium. This result is found to hold regardless of the
strategy the sophisticated firms adopt. It was further demonstrated in [3, 4] that either in dynamical transitionary periods or when the
economy turns cyclic or chaotic, a combination of the price-taking strategy
with a simple cautious adjustment strategy can also lead to relatively higher
average profits for a firm than its rival, if the latter adopts a myopic
Cournot best-response strategy.The current research follows the same trend of
research and explores the relative performance of price-taking strategy in
terms of profit and sales revenue when a novel adjustment strategy—the
growth-rate adjustment strategy—is implemented to prevent the economy
collapse, or to force the market price to stay in an economically meaningful
region. Using a traditional linear economy with linear demand and marginal
cost, we are able to show in theory and demonstrate by numerical simulations
that by combining the price-taking strategy with growth-rate adjustment
strategy, a firm commands both an unbeatable advantage in terms of the relative
performance (such as profit and sales revenue) as well as the stabilization
power in a dynamically unstable market.The article is organized as follows. A heterogeneous
oligopoly model is set up in Section2 and the relevant dynamical
characteristics are summarized in Section 3. Section 4 examines the relative
performance measures in equilibrium and their implications in general. The
analysis for the relative performance in disequilibrium
is presented in Section 5. Section 6 concludes the
research.


## 2. A Heterogeneous Oligopoly Model

Consider an oligopoly market, in whichN=1+m firms produce a homogeneous product with
quantity qti,i=1,2,…,1+m,
at period t.
The inverse market demand for the product is given by pt=D(Qt),
where D′(⋅)≤0,
with equality holding only at finite number of points. The conventional
assumption that Qt=∑i=1Nqti,
that is,
the actual market price adjusts to the demand so as to clear
the market at every period, applies.While all firms are assumed to have an identical
technology and hence an identical convex cost functionC, with C′>0 and C′′>0,
they adopt different production strategies. We assume that all firms can be
classified into two categories: theprice-takers and theCounortoptimizers.The first firm is the price-taker, who is either
deficient in market information or less strategic in market competition. Its
production target, denoted byx^t is derived by equating the marginal cost
(i.e., the first-order derivative of cost) with market price if last period,
that is, C′(x^t)=pt−1.
Notice that pt−1=D(Qt−1),
the production target x^t can be expressed as(2.1)x^t=MC−1(D(Qt−1)),where MC−1 denotes the inverse function of C′.Instead of producing atx^t,
the price-taker is assumed to implement the growth-rate adjustment strategy so
as to limit the production growth rate to a positive constant γ in the sense that its output, denoted by xt,
is determined from(2.2)xt−xt−1xt−1=γ(x^t−xt−1),or, equivalently,
from(2.3)xt=xt−1(1+γ(MC−1(D(Qt−1))−xt−1)).Remark 2.1.
This mechanism was proposed in [2]
as one of the adaptive adjustment mechanisms purely for the purpose of
stabilizing unstable dynamics or controlling chaos. Compared to other
approaches, the mechanism inherits two unique advantages of feedback-adjustment
type of methodologies: (i) requiring neither prior information about the system
itself nor any externally generated control signal(s); (ii) forcing the adjusted
system to converge to generic periodic points. Moreover, it is very effective
and efficient in implementing and achieving the goal of
stabilization.In contrast, the restm firms are Counortoptimizers.
They know exactly the price-taker's output. To take this information advantage
and the market power to maximum extent, they form a collusion and
produce an identical quantity yt to maximize the individual profit given
by(2.4)πy(xt,yt)=sy(xt,yt)−C(yt)=D(Qt)yt−C(yt),where sy(xt,yt)=ptyt=D(Qt)yt is the sales revenue.Then the first-order profit maximization condition∂πy/∂yt=0 yields(2.5)D(Qt)+∂D(Qt)∂ytyt−C′(yt)=0.SinceQt=xt+myt,
we have ∂D(Qt)/∂yt=mD′(Qt) and so that (2.5) is simplified to (2.6)D(Qt)+mytD′(Qt)=C′(yt).Assume that appropriate second-order condition is
satisfied so thatyt=Ry(xt) implicitly solved from (2.6) leads to a profit
maximum.For the convenience of reference, we will call the
dynamical process given by (2.3) and (2.6) together as a general
heterogeneous
oligopoly model (GHO model). To avoid running into conclusions that are
sensible in mathematics but meaningless in economics, we will focus on the
scenarios that only bring economical meaningful solutions. To this end, an
economically meaningful concept needs to be defined formally.Deffinition.
An output bundle(xt,yt) for the generalheterogeneous
oligopoly model is said to be economicallymeaningful, if the
following inequalities are met:
(i)
xt≥0 and yt≥0,
but qt>0,
that is, nonnegative market supply;(ii)
0<pt=D(qt)<∞, that is, positive and limited price.


## 3. Dynamical Analysis of a Linear Model

To have a deeper understanding of the role of the
price-taker in our model, a concrete example that can be manipulated
analytically and verified is indispensable. For this purpose, we will illustrate
our main points with alinearheterogeneous oligopoly model (LHO
model), in which the market demand is assumed to be linear so that its inverse
demand function is given by(3.1)pt=D(xt+myt)=1−xt−myt,whereas the marginal cost is
linear so that the cost function
adopts the form of(3.2)C(q)=C0+c2q,where both C0 and c are positive constants. Without loss of
generality,  we assume that C0=0.Substituting (3.1) and (3.2) into (2.3)
yields(3.3)xt=xt−1(1+γ((1−xt−1−myt−1)c−xt−1)).On the other hand, (2.6) defines an optimal reaction
function for the Cournot optimizers:(3.4)yt=Ry(xt)=1−xt2m+c.Substituting (3.4) into (3.3) and manipulating
them yield a quadratic recurrence
relation(3.5)xt=f(xt−1)=˙a(γ)xt−1(1−b(γ)xt−1),where(3.6)a(γ)=˙1+γ(c+m)c(c+2m),b(γ)=˙γc+m+c(c+2m)(c+m)γ+c(c+2m).For the discrete process (3.5), the trajectory {xt}t=0 is bounded by x0*(γ)=1/b(γ) if and only if 0≤x0≤x0*(γ) and a(γ)≤4.
However, to generate an economically meaningful trajectory {xt}t=0,
we need also ensure that x0<1,
which is guaranteed if we have x0*(γ)<1 and a(γ)≤4,
or equivalently, if(3.7)1<γ≤γmax⁡,where(3.8)γmax⁡=3c(c+2m)c+m.Figure1 illustrates the impact of γ on the functional shape of f for the case of c=1.
We see that(3.9)∂x0*(γ)∂γ=−cγ2c+2mc+m+c(c+2m)<0,∂x0*(γ)∂m=c2γ1−γ(c+m+2cm+c2)2<0ifγ>1.Illustration ofxt=f(xt−1),
relative profitability regime Ωp and the trapping set Ω(γ).
(a)
m=3(b)
m=1The dynamic characteristics of (3.5) have been studied
substantially in literature and presented in many textbooks. So we will
summarize some well-known results directly without repeating the
details.
Theorem 3.1.
For the discrete dynamic process (3.5),
when inequality (3.7) is satisfied, one has the following:
(i)
there exists a unique nontrivial equilibriumx¯:(3.10)x¯=m+cm+c+2mc+c2,at which the derivative of f is given by(3.11)σ=˙df(x)dx|x¯=1−γ(c+m)c(c+2m);(ii)
the equilibriumx¯ is stable for γ<γ1,
and the converge to the equilibrium is monotonic if γ<γ0 and cyclical if γ>γ0,
where(3.12)γ0=c(c+2m)(c+m),(3.13)γ1=2γ0;(iii)
increasingγ from γ1 to γ∞,
where(3.14)γ∞≈2.5699⁢γ0will lead to a sequence of
period-doubling bifurcations, that is, stable cycles with periods 2n appear right after the cycles with periods 2n−1 become unstable, for n≥1;(iv)
increasingγ from γ∞ to γmax,
stable cycles for all periods can appear at some specific intervals for γ while aperiodic oscillations occur for the
rest value of γ;(v)
forγ=γmax,
no stable periodic-cycles exist for any order and the system is pure chaotic in
the topological sense, that is, sensitive
dependence on the initial condition, and the probabilistic sense, that is, the
frequency from the random-like trajectories is however independent of the
initial conditions and obeys an invariant densityφthat is absolutely continuous with respect to
the Lebesgue measure, where(3.15)φf(x)=1πx(x¯max⁡−x)forx∈J=˙[0,x¯max⁡],where(3.16)x¯max⁡=x0*(γmax⁡)=4(c+m)3(c+m+2cm+c2);(vi)
forγ>γmax,
the trajectory starting from almost all x0>0 becomes unbounded and hence is not
economically meaningful.Proof.
At first, we point out that the mapf is always topologically conjugated with the
logistic map, via a linear homeomorphism. Condition (i) can be directly
verified. Conditions (ii)–(iv) and (vi) follow from the standard textbook
analysis.
(v) Whenγ=γmax,f is topologically conjugated
with(3.17)Xt=F(xt−1)⁢=˙4Xt−1(1−Xt−1)in the sense that f=h−1∘F∘h,
where h:J→[0,1] is a homeomorphism defined by(3.18)h(x)=xx¯max⁡,forx∈J.
By the law of conjugation, ifφF(X) is the invariant density for F,
then the invariant density for f is given by(3.19)φf(x)=φF(h(x))|dhdx|.
It is well known that the invariant density preserved
by (3.17) is given by(3.20)φF(X)=1πX(1−X)forX∈[0,1].
Substituting withh defined in (3.18) into (3.19) results in (3.15).Remark 3.2.
Details about the chaos
in probabilistic sense and relevant issues of statistic dynamics can be found
in [5, 6]. Chaos in probabilistic sense occurs
for infinitely many other values in the compact interval [γ∞,γmax] as well. However, the invariant density in
general cannot be expressed as the composition of finite terms of basic
functions.Remark 3.3.
While the value of the nontrivial
steady-statex¯ is independent of the adjustment rate γ,
the stability does depend on γ.
For a fixed γ>γ0,
the stability of x¯ is characterized by the derivative given by
(3.11), from which we have(3.21)∂σ∂c=γ(c+m)2+m2c2(c+2m)2>0,∂σ∂m=γ(c+2m)2>0.Thus, increasing number of
Cournot optimizers stabilizes the economy, but increasing the technological
level (less c) destabilizes the economy.Figures2(a) and 2(b) depict the bifurcation
diagram with respect to γ and c,
respectively. A structural similarity is apparently observed between them.Bifurcations ofxt=f(xt−1),m=3.
(a)
Bifurcation with respect toγ(c=1/2)(b)
Bifurcation with respect toc(γ=2.5)


## 4. Relative Performance in Equilibrium

To see the merits of (3.3), we will proceed to evaluate
the relative performance between the price-taker and the Cournot optimizer. The profits and sales revenue as two of most important performance
measures used in the business and accounting literature will be
considered. However, sales revenue is taken as auxiliary index evaluated from the
price-taker's point of view instead of from the collusion's point of view since
the latter is assumed to be a profit-maximizer as implied by the optimal
reaction given by (2.6).
Unlike being treated in [7],
we do not evaluate these indices in the aggregated sense because the long-run
averages of the relevant performance measures provide equivalent information
both in cyclical and chaotic market environment.


### 4.1. Profits

At each period, the price is given by(4.1)pt=1−xt−myt=m+c2m+c(1−xt),that is, the realized price is
in fact negatively related to the price-taker's output. However, compared to
the case in which there is no counter-reaction from the Cournot optimizers, the
sensitivity is reduced from unity to a fraction (since (m+c)/(2m+c)<1).The profits made by the price-taker and
by any of the Cournot optimizers
are(4.2)πtx=(pt−cxt2)xt=m+c2m+c(1−2(m+c)+c(c+2m)2(m+c)xt)xt,πty=(pt−cyt2)yt=(m+c2)yt2=(1−xt)22(2m+c),respectively.We see that,πty≥0 for all t,
that is, the Cournot optimizers make no profit loss at each and every move due
to its optimization strategy. (This result is
actually independent of the specific demand function and the cost function. It
can be shown that it is a typical characteristic of a firm that adopts
Cournot-type strategy with full-information about its rival's outputs when the
fixed cost is negligible.) On the contrary, the
price-taker may suffer from profit loss due to its lagged reaction to the
market changes. In fact, πtx>0 if and only if xt<x+,
where x+ is defined by(4.3)x+=2(m+c)2(m+c)+c(2m+c)<1.Simple mathematical manipulations indicate
that(4.4)Δπtxy=˙πtx−πty=(c+2m+1)(1+c)2(2m+c)(x*−xt)(xt−x*),where x*=˙1/(1+c+2m) and x*=˙1/(1+c).Therefore,πtx≥πty if and only if xt∈Ωp=˙[x*,x*].Ωp so defined will be referred to as the relative
profitability regime for the price-taker. It is worthwhile to note that the relative profitability regime is independent of the adjustment rate, γ,
which enables a firm to adjust the trapping set (to be discussed in Section 5)
of the trajectory so as to maintain the relative profitability advantage over
the Cournot optimizers.


### 4.2. Sales Revenue

In many business competition, sales revenue is a
benchmark for market share. In our settings, they are given by(4.5)stx=ptxt=m+c2m+cxt(1−xt),sty=ptyt=m+c(2m+c)2(1−xt)2.We see thatstx and sty are always positive (due to the fact that xt<1 is implicitly guaranteed). Moreover,
since(4.6)Δstxy=˙stx−sty=pt(xt−yt), we have Δstxy>0 so long as xt>yt,
which occurs when (4.7)xt>x*=1(1+c+2m),that is, when the price-taker's
output exceeds the lower bound of the relative profitability regime Ωp.


### 4.3. Relative Performance

At the unique nontrivial equilibriumx¯ given by (3.10), the other relevant equilibrium
values are summarized in the second column of Table 1. It can be verified that x¯∈Ωx,
regardless of the values of c, m,
and γ.
Therefore, at the equilibrium, the price-taker enjoys a higher sale revenue
as well as a higher profit than the Cournot optimizers.Table 1
Equilibrium outcomes.
All price-taker1 price-takerm ColludedAll ColludedPricep¯w=c1+m+cp¯=c(m+c)2cm+c2+m+cp¯u=m+c+12m+2+cOutputsq¯w=11+m+cx¯=m+c2cm+c2+m+cy¯=c2cm+c2+m+cq¯u=12m+2+cProfitsπ¯w=c2(1+m+c)2π¯x=12c(m+c)2(2cm+c2+m+c)2π¯y=12c2(2m+c)(2cm+c2+m+c)2π¯u=12(2+2m+c)Saless¯w=c(1+m+c)2s¯x=c(m+c)2(2cm+c2+m+c)2s¯y=c2(m+c)(2cm+c2+m+c)2s¯u=m+c+1(2m+2+c)2For benchmarks, two extreme situations are also
included in Table1, one with all m+1 firms adopting the price-taking strategy and
the other with all m+1 firms forming a collusion.In the former case where all firms behave as the
price-taker, the equilibrium is known as aWalrasian equilibrium (competitive equilibrium). For the latter case, all firms behave as if they
were a monopolist and maximize the total profit with a constant
output(4.8)qu=12m+2+c,from which the other performance
measure can be evaluated. (Again we do not consider
the case in which the monopoly maximizes the sales
revenue.)While Figure3 provides graphical illustrations of the
outputs, the sales revenue, and the profits at the equilibrium with m=1 and m=3.Critical outputs and performance measures.
(a)
m=1(b)
m=3In fact, the conclusions for the equilibrium analysis
are generic in the sense that they are independent of the concrete functional
forms of the linear demand and the quadratic cost functions illustrated in our
analysis.Theorem 4.1.
Consider the GHO model given by(4.9)xt=xt−1(1+γ(MC−1(D(Qt))−xt−1)),(4.10)C′(yt)=D(Qt)+mytD′(Qt),where Qt=˙xt+myt,
the cost function is strictly convex (C′>0 and C′′>0) and the demand function (D′<0) satisfies the additional assumptions that both D(Qt) and D′(Qt) are bounded for Qt≠0,
the following facts hold.
(i)
All nontrivial intertemporal equilibria are
economically meaningful.(ii)
x¯>y¯>0 so that s¯x>s¯y>0.(iii)
π¯x>π¯y.
Moreover, π¯x>π¯y>0 if C(0)=0.Proof.
(i)-(ii)
At a nontrivial intertemporal equilibrium (steady-states(x¯,y¯)), we mean x¯>0.
It follows from (4.9) that we must have(4.11)p¯=C′(x¯).
The facts thatC′′>0 thus implies p¯>0.
What remains to be verified is the positiveness of y¯.
Equation (4.10) yields(4.12)C′(x¯)+mD′(Q¯)y¯=C′(y¯),where Q¯=x¯+my¯.
Since D′(Q¯) is bounded even when y¯=0,
a hypothesis that y¯=0 would lead to(4.13)C′(x¯)=C′(y¯)=C′(0).However, (4.13) in turn suggests
that x¯=0 by the monotonicity characteristics implied
from the convexity assumption of C,
which is a contradiction to the fact that x¯>0.
Again from (4.12), a positive y¯ and D′<0 together thus suggest x¯>y¯.
Under the same equilibrium price p¯, we thus have s¯x>s¯y>0.
(iii) For the profit difference at the nontrivial
equilibrium, (4.11) suggests that(4.14)Δπ¯xy=π¯x−π¯y=C′(x¯)(x¯−y¯)−(C(x¯)−C(y¯)).
It follows from the assumption ofC′′(⋅)>0 and x¯≠y¯ that C′(x¯)(x¯−y¯)−(C(x¯)−C(y¯))>0,
or equivalently,(4.15)π¯x>π¯y.
Next we need to show thatπ¯y>0 if the fixed cost is negligible. It follows
from (4.12) that p¯=C′(y¯)−mD′(Q¯)y¯ so that(4.16)π¯y=p¯⁢y¯−C(y¯)=(C′(y¯)−mD′(Q¯)y¯)y¯−C(y¯)=C′(y¯)y¯−C(y¯)+(−mD′(Q¯)y¯2),which is positive since C′(y¯)y¯>C(y¯) if C(0)=0.Remark 4.2.
Due to the limit scope, theoretical issues such as
the existence of nontrivial equilibrium and the selection mechanism when
multiple nontrivial equilibria are presented are not discussed since
we are only concerned with with the situations that can
lead to economically meaningful outcomes, based on which the relative performance
of two types of firms can be evaluated. Nevertheless, it can be proved that a
unique nontrivial equilibrium exists when a classical assumption about the
marginal revenue as justified in [8, 9] is assumed.Remark 4.3.
In the proof of part (iii), we do
not use the fact that the outputs of the Cournot optimizers follow (4.10), thus
the conclusion for the relative profitability advantage of the price-taker
holds for more general situations where each Cournot optimizer adopts different
strategy (including partial and full collusion), provided that its equilibrium
output, y¯k,k=1,2,…,m,
is different from x¯.
Moreover, the assumption for the strict convexity can be relaxed to convexity
so long as the marginal cost is not a constant.Remark 4.4.
The relative profitability in an equilibrium for the price-taker, though
seemingly contradictory to economic intuition, is still justified by economic
theory. In fact, at an equilibrium, the market price approaches to a constant
and does not respond to the variation of individual output. As a result, equating marginal cost to the market
price is the optimal decision.An equilibrium may exist in theory but may never be
reached (converged to) in reality. Whenγ is sufficiently large (γ>γ∞ in LHO model), the asymptotic behavior becomes
very complex, ranging from stable periodic cycles to aperiodic cyclic
fluctuations, as well as the situations in which a stable cycle coexists with
infinitively many unstable cycles. Therefore, to complete the analysis of
relative performance, we need to investigate what happens when the trivial
equilibrium becomes unstable, which leads us to the next section.


## 4.1. Profits

At each period, the price is given by(4.1)pt=1−xt−myt=m+c2m+c(1−xt),that is, the realized price is
in fact negatively related to the price-taker's output. However, compared to
the case in which there is no counter-reaction from the Cournot optimizers, the
sensitivity is reduced from unity to a fraction (since (m+c)/(2m+c)<1).The profits made by the price-taker and
by any of the Cournot optimizers
are(4.2)πtx=(pt−cxt2)xt=m+c2m+c(1−2(m+c)+c(c+2m)2(m+c)xt)xt,πty=(pt−cyt2)yt=(m+c2)yt2=(1−xt)22(2m+c),respectively.We see that,πty≥0 for all t,
that is, the Cournot optimizers make no profit loss at each and every move due
to its optimization strategy. (This result is
actually independent of the specific demand function and the cost function. It
can be shown that it is a typical characteristic of a firm that adopts
Cournot-type strategy with full-information about its rival's outputs when the
fixed cost is negligible.) On the contrary, the
price-taker may suffer from profit loss due to its lagged reaction to the
market changes. In fact, πtx>0 if and only if xt<x+,
where x+ is defined by(4.3)x+=2(m+c)2(m+c)+c(2m+c)<1.Simple mathematical manipulations indicate
that(4.4)Δπtxy=˙πtx−πty=(c+2m+1)(1+c)2(2m+c)(x*−xt)(xt−x*),where x*=˙1/(1+c+2m) and x*=˙1/(1+c).Therefore,πtx≥πty if and only if xt∈Ωp=˙[x*,x*].Ωp so defined will be referred to as the relative
profitability regime for the price-taker. It is worthwhile to note that the relative profitability regime is independent of the adjustment rate, γ,
which enables a firm to adjust the trapping set (to be discussed in Section 5)
of the trajectory so as to maintain the relative profitability advantage over
the Cournot optimizers.


## 4.2. Sales Revenue

In many business competition, sales revenue is a
benchmark for market share. In our settings, they are given by(4.5)stx=ptxt=m+c2m+cxt(1−xt),sty=ptyt=m+c(2m+c)2(1−xt)2.We see thatstx and sty are always positive (due to the fact that xt<1 is implicitly guaranteed). Moreover,
since(4.6)Δstxy=˙stx−sty=pt(xt−yt), we have Δstxy>0 so long as xt>yt,
which occurs when (4.7)xt>x*=1(1+c+2m),that is, when the price-taker's
output exceeds the lower bound of the relative profitability regime Ωp.


## 4.3. Relative Performance

At the unique nontrivial equilibriumx¯ given by (3.10), the other relevant equilibrium
values are summarized in the second column of Table 1. It can be verified that x¯∈Ωx,
regardless of the values of c, m,
and γ.
Therefore, at the equilibrium, the price-taker enjoys a higher sale revenue
as well as a higher profit than the Cournot optimizers.Table 1
Equilibrium outcomes.
All price-taker1 price-takerm ColludedAll ColludedPricep¯w=c1+m+cp¯=c(m+c)2cm+c2+m+cp¯u=m+c+12m+2+cOutputsq¯w=11+m+cx¯=m+c2cm+c2+m+cy¯=c2cm+c2+m+cq¯u=12m+2+cProfitsπ¯w=c2(1+m+c)2π¯x=12c(m+c)2(2cm+c2+m+c)2π¯y=12c2(2m+c)(2cm+c2+m+c)2π¯u=12(2+2m+c)Saless¯w=c(1+m+c)2s¯x=c(m+c)2(2cm+c2+m+c)2s¯y=c2(m+c)(2cm+c2+m+c)2s¯u=m+c+1(2m+2+c)2For benchmarks, two extreme situations are also
included in Table1, one with all m+1 firms adopting the price-taking strategy and
the other with all m+1 firms forming a collusion.In the former case where all firms behave as the
price-taker, the equilibrium is known as aWalrasian equilibrium (competitive equilibrium). For the latter case, all firms behave as if they
were a monopolist and maximize the total profit with a constant
output(4.8)qu=12m+2+c,from which the other performance
measure can be evaluated. (Again we do not consider
the case in which the monopoly maximizes the sales
revenue.)While Figure3 provides graphical illustrations of the
outputs, the sales revenue, and the profits at the equilibrium with m=1 and m=3.Critical outputs and performance measures.
(a)
m=1(b)
m=3In fact, the conclusions for the equilibrium analysis
are generic in the sense that they are independent of the concrete functional
forms of the linear demand and the quadratic cost functions illustrated in our
analysis.Theorem 4.1.
Consider the GHO model given by(4.9)xt=xt−1(1+γ(MC−1(D(Qt))−xt−1)),(4.10)C′(yt)=D(Qt)+mytD′(Qt),where Qt=˙xt+myt,
the cost function is strictly convex (C′>0 and C′′>0) and the demand function (D′<0) satisfies the additional assumptions that both D(Qt) and D′(Qt) are bounded for Qt≠0,
the following facts hold.
(i)
All nontrivial intertemporal equilibria are
economically meaningful.(ii)
x¯>y¯>0 so that s¯x>s¯y>0.(iii)
π¯x>π¯y.
Moreover, π¯x>π¯y>0 if C(0)=0.Proof.
(i)-(ii)
At a nontrivial intertemporal equilibrium (steady-states(x¯,y¯)), we mean x¯>0.
It follows from (4.9) that we must have(4.11)p¯=C′(x¯).
The facts thatC′′>0 thus implies p¯>0.
What remains to be verified is the positiveness of y¯.
Equation (4.10) yields(4.12)C′(x¯)+mD′(Q¯)y¯=C′(y¯),where Q¯=x¯+my¯.
Since D′(Q¯) is bounded even when y¯=0,
a hypothesis that y¯=0 would lead to(4.13)C′(x¯)=C′(y¯)=C′(0).However, (4.13) in turn suggests
that x¯=0 by the monotonicity characteristics implied
from the convexity assumption of C,
which is a contradiction to the fact that x¯>0.
Again from (4.12), a positive y¯ and D′<0 together thus suggest x¯>y¯.
Under the same equilibrium price p¯, we thus have s¯x>s¯y>0.
(iii) For the profit difference at the nontrivial
equilibrium, (4.11) suggests that(4.14)Δπ¯xy=π¯x−π¯y=C′(x¯)(x¯−y¯)−(C(x¯)−C(y¯)).
It follows from the assumption ofC′′(⋅)>0 and x¯≠y¯ that C′(x¯)(x¯−y¯)−(C(x¯)−C(y¯))>0,
or equivalently,(4.15)π¯x>π¯y.
Next we need to show thatπ¯y>0 if the fixed cost is negligible. It follows
from (4.12) that p¯=C′(y¯)−mD′(Q¯)y¯ so that(4.16)π¯y=p¯⁢y¯−C(y¯)=(C′(y¯)−mD′(Q¯)y¯)y¯−C(y¯)=C′(y¯)y¯−C(y¯)+(−mD′(Q¯)y¯2),which is positive since C′(y¯)y¯>C(y¯) if C(0)=0.Remark 4.2.
Due to the limit scope, theoretical issues such as
the existence of nontrivial equilibrium and the selection mechanism when
multiple nontrivial equilibria are presented are not discussed since
we are only concerned with with the situations that can
lead to economically meaningful outcomes, based on which the relative performance
of two types of firms can be evaluated. Nevertheless, it can be proved that a
unique nontrivial equilibrium exists when a classical assumption about the
marginal revenue as justified in [8, 9] is assumed.Remark 4.3.
In the proof of part (iii), we do
not use the fact that the outputs of the Cournot optimizers follow (4.10), thus
the conclusion for the relative profitability advantage of the price-taker
holds for more general situations where each Cournot optimizer adopts different
strategy (including partial and full collusion), provided that its equilibrium
output, y¯k,k=1,2,…,m,
is different from x¯.
Moreover, the assumption for the strict convexity can be relaxed to convexity
so long as the marginal cost is not a constant.Remark 4.4.
The relative profitability in an equilibrium for the price-taker, though
seemingly contradictory to economic intuition, is still justified by economic
theory. In fact, at an equilibrium, the market price approaches to a constant
and does not respond to the variation of individual output. As a result, equating marginal cost to the market
price is the optimal decision.An equilibrium may exist in theory but may never be
reached (converged to) in reality. Whenγ is sufficiently large (γ>γ∞ in LHO model), the asymptotic behavior becomes
very complex, ranging from stable periodic cycles to aperiodic cyclic
fluctuations, as well as the situations in which a stable cycle coexists with
infinitively many unstable cycles. Therefore, to complete the analysis of
relative performance, we need to investigate what happens when the trivial
equilibrium becomes unstable, which leads us to the next section.


## 5. Relative Performance in Disequilibrium

Due to complex nature of the nonlinear dynamics for an
unstable GHO model, we need to resort to a concrete example like LHO to provide
a clearer picture.As indicated in Section3, when γ>γ1,
stable and unstable periodic cycles as well as aperiodic cycles may appear. In
these cases, we are interested in comparing the long-run average performance of
the price-taker and the Cournot optimizers. A formal definition for the average
is needed.
Definition 5.1.
LetG∈C1 be a performance measure function of xt.
Given an adjustment speed γ and an associated periodic-k cycle {x¯i(γ)}i=1k,
its k-period average is defined
as(5.1)〈Gk〉γ=˙1k∑i=1kG(x¯i(γ)),and its long-run average for a trajectory {xt}t=0∞ is defined as(5.2)〈G〉γ=˙lim⁡T→∞1T∑t=0TG(xt).For the discrete dynamic process (3.5), if {x¯i(γ)}i=1k is a stable periodic cycle, then we
have(5.3)〈G〉γ=〈Gk〉γ.Apparently, both the performance measure functions:
profits and sales revenue, are continuous at least
up to the first-order differential, so are the
respective differences between the price-taker and the Cournot optimizer.Before proceeding to evaluate the periodic-k average and the long-run average of these
measures in the relative terms, we have to make sure that they are economically
meaningful.
Theorem 5.2.
For the LHO modelgiven by (3.5)
and (3.4), for γ∈(0,γmax),
where γmax is defined in (3.8), one
always has(5.4)〈Gkx〉γ>0,〈Gky〉γ>0foranyk≥1,and more
generally,(5.5)〈Gx〉γ>0,〈Gy〉γ>0,where G denotes π and s.Proof.
The conclusion for the sales revenues follows straightforwardly from the fact that xt>0 for all t if γ∈(0,γmax) and x0<x0*(γ),
as discussed in Section 3.
For the profitπ,
we note first that (4.9) implies that(5.6)1γ(xt+1xt−1)+xt=MC−1(pt),that is,(5.7)pt=cγ(xt+1xt−1)+cxt,which leads to(5.8)πx(xt)=ptxt−C(xt)=cγ(xt+1−xt)+c2xt2.
For a given periodic-k cycle {x¯i}i=1k,
we have(5.9)〈πkx〉γ=˙1k∑i=1kπx(x¯i)=1k∑i=1k((c/γ)(x¯i+1−x¯i)+(c/2)x¯i2)=ckγ(x¯k+1−x¯1)+c2k∑i=1kx¯i2=c2k∑i=1kx¯i2>0,where we  utilize the fact that x¯k+1=x¯1.
Similarly, for any chaotical trajectory{xt}t=1∞,
the long-run average of profit for the price-taker can be recast
as(5.10)〈πx〉γ=˙lim⁡T→∞⁢1T∑t=1Tπx(xt)=cγ⁢lim⁡T→∞⁢1T(xT−x1)+c2⁢lim⁡T→∞⁢1T∑i=1Txi2=c2⁢lim⁡T→∞⁢1T∑i=1Txi2>0,which holds because the limit
term limT→∞(1/T)(xT−x1) approaches zero since both xT and x1 are less than unity by assumption.Next, we turn to explore the relative performance
between two types of firms in disequilibrium. Since the requirements for the
relative performance in terms of sales-revenue are weaker than those in terms
of profits, we will focus our discussion more on the latter.From the analysis in Section4, we see that there
exists a relative profitability regime Ωp that is independent of the adjustment rate γ such that the price-taker will make more
profit than the Cournot optimizer whenever the former's output falls in that
regime.On the other hand, as illustrated in Figure1, with
fixed m and c,
for γ0<γ≤γmax,
where γ0 and γmax are, respectively, defined in (3.12) and (3.8),
there exists a trapping set Ω(γ)=˙[xmin(γ),xmax(γ)] such that starting from any initial output x0∈(0,x¯0(γ)),
the output trajectories {xt}t=1∞ will be eventually trapped in Ω(γ) and remain there forever. xmin(γ) and xmax(γ) are determined from(5.11)xmax⁡(γ)=f(x0*(γ)2)=(γ(c+m)+c(c+2m))24cγ(c+2m)(c+m+c(c+2m)),xmin⁡(γ)=f(xmax⁡(γ))=(γ(c+m)+c(c+2m))3(3c(c+2m)−γ(c+m))16c3γ(c+2m)3(c+m+c(c+2m)).Forγ<γmax,xmin(γ)>0. Moreover, ∂xmax(γ)/∂γ>0 and ∂xmin(γ)/∂γ<0 for γ0≤γ≤γmax.From the fact thatxmin(γ0)=xmax(γ0)=x¯,
we can infer that the trapping set Ω(γ) grows around x¯ along with increasing value of γ.
This in turn suggests that for a sufficiently small positive ϵ and γ=γ0+ϵ,
we are able to enforce Ω(γ)⊂Ωp.
That is to say, by limiting the magnitude of γ,
the price-taker is able to secure that the trapping set falls within the
relative profitability regime so as to achieve a relative advantage in terms of
profit. Similar reasoning can be applied to the analysis of the sales revenue.Since the condition forstx>sty is just xt>x*,
which can be guaranteed as long as the lower-bound of Ω(γ) coincides with the lower-bound of Ωp,
solving the identity(5.12)xmin⁡(γ*s)=x*yields an upper bound γ*s for the adjustment rate to ensure stx>sty for all xt∈Ω(γ).Denote byγ* the maximum possible γ such that the upper bound of the trapping set Ω(γ) coincides with the upper bound of the relative
profitability regime Ωp,
then γ* is a solution to the identity xmax(γ*)=x*.Finally, letγ*p=˙min{γ*s,γ*},
then we have γ*s≥γ*p by definition and the following beautiful
results.Theorem 5.3.
With the LHO model defined in (3.4)
and (3.5), for any given m and c,
(i)
whenγ∈[γ0,γ*p],
one has  πtx>πty>0 for all xt∈Ω(γ),
that is, the price-taker makes more profit than the individual Cournot optimizer
at each move after the trajectory runs into the trapping set;(ii)
whenγ∈[γ0,γ*s],
one has  stx>sty>0 for all xt∈Ω(γ),
that is, the price-taker has higher sales revenue than the individual Cournot
optimizer at each move after the trajectory runs into the trapping set.Table2 provides a list of γ*s and γ*p values for several typical sets of parameters
we have applied in our illustrations.Table 2
Some typical values.
γ*pγ*sc=1/2,m=12.07232.0955c=1/2,m=32.58482.5898c=1,m=13.57773.5777c=1,m=34.73874.7387Whenγ exceeds the upper bounds γ*p and γ*s,
respectively, Δπtxy and ΔStxy become negative from time to time. Then we are
again forced to consider periodic-k average and the long-run averages of the
performance measures.In terms of profit, it follows from Theorem5.3 that 〈Δπxy〉γ>0 is guaranteed for all xt∈Ω(γ),
if γ∈[γ0,γ*p].
By the continuity of πtx and πty as functions of γ and the ergodicity of the dynamical process
(3.5), we can hypothesize that there exists a γ¯kp such that so long as γ0≤γ<γ¯kp,
despite that Δπtxy<0 for some t,
we still have 〈Δπkxy〉γ>0 for any cyclic orbit. Apparently, the value of γ¯kp depends on the order k.
For instance, for a period-2 cycle (x¯1(γ),x¯2(γ)), with x¯1(γ)≤x¯2(γ),
(4.4) suggests that(5.13)〈Δπ2xy〉γ=12(Δπxy(x¯1(γ))+Δπxy(x¯2(γ)))⋈12∑i=12(x*−x¯i(γ))(x¯i(γ)−x*).The RHS of (5.13) is a continuous
function of γ.
In considering the fact that x¯1(γ0)=x¯2(γ0)=x¯, where γ0 is defined in (3.12), and the differential
properties ∂x¯1(γ)/∂γ>0 and ∂x¯2(γ)/∂γ<0 for γ1≤γ≤γmax,
we are able to conclude the existence of a γ¯2p, γ¯2p>γ0,
such that the equality 〈Δπxy(γ)〉2=0 holds.Similar conclusion can be expected for an aperiodic
orbit which occurs whenγ>γ∞.
There exists a γ¯p value such that for all γ<γ¯p,
we have 〈Δπxy〉γ>0.
There is no analytical formula to obtain the value of γ¯p.Figures4 to 6 depict the numerical simulations of
relevant performance measures for several combinations of m and c values.Long-run averages with respect toγ(m=3,c=1/2).
(a)
Profit(b)
Sales revenueComparing Figure4(a) with Figure 5(a), we see that γ¯p decreases with the increase in c.
For a relatively small c (a more divergent economy caused by higher
marginal cost), γ¯p can even exceed its maximum possible
growth-rate limit γmax,
as illustrated in Figure 4. On the other hand, by comparing Figure 4(a) with
Figure 6(a), we see that increasing m (more Cournot optimizers) stabilizes the
economy so that γ¯p increases along with the increase of the
maximum possible growth-rate limit γmax.Long-run averages with respect toγ(m=3,c=1).
(a)
Profit(b)
Sales revenueLong-run averages with respect toγ(m=1,c=1/2).
(a)
Profit(b)
Sales revenueSimilarly, there exist cases in which〈sx〉γ>〈sy〉γ for all γ<γmax,
as illustrated in Figures 4(b) and 5(b).The above analysis and observations can be summarized
into the following theorem.Theorem 5.4.
With the LHO model defined in (3.4)
and (3.5), for any given m and c,
(i)
there existsγ¯p,γ¯p>γ*p,
such that for all γ∈[γ0,γ¯p],
one has  〈Δπxy〉γ>0,
that is, the long-run average profits made by the price-taker is higher than
that earned by each individual Cournot optimizer;(ii)
there existsγ¯s,γ¯s≥γ*s,
such that for all γ∈[γ0,γ¯s],
one has  〈Δsxy〉γ>0,
that is, the long-run average sales-revenue earned by the price-taker is higher
than that earned by each individual Cournot optimizer.Whenγ=γmax,
process (3.5) becomes full-chaotic and preserves an ergodic invariant density φf given by (3.15) so that for any function G∈C1,
its space mean equals to the time mean in the sense of(5.14)〈G〉γ=∫0x¯max⁡G(x)φf(x)dx.For instance, substitutingΔπtxy defined in (4.4) into (5.14) leads
to(5.15)〈Δπxy〉γmax⁡=∫0x¯max⁡Δπxy(x)φf(x)dx⋈∫0x¯max⁡(x*−x)(x−x*)φf(x)dx⋈∫0x¯max⁡(x*−x)(x−x*)x(x¯max⁡−x)dx.Asx*,x*,
and x¯max all depend on m and c,
so does the integral in (5.15). Analogously, all long-run averages of performance
measures under full chaos can be numerically evaluated. Figure 7 provides such
results with respect to c for m=3,
from which the impact of c is illustrated. Again, as expected, a range
for γ can be identified, in which we have 〈Δπxy〉γmax>0.Performance
measures under full-chaos (γ=γmax,m=3).
(a)
Profit(b)
Sales revenueAlthough our analysis is conducted using the LHO
model, the methodology applied and conclusions drawn in Theorem5.4 are expected
to hold for the GHO model in general. Conclusive results cannot be obtained
without adequate analytical assumptions on the derivative properties of cost
function and demand in general, and hence will be presented in the future
report.


## 6. Conclusions

A heterogeneous oligopoly model consisting of a
price-taking firm and a group of Cournot optimizers have been studied from both
equilibrium and disequilibrium points of view. If the Cournot optimizers form a
collusion and maximize their profit with the full information about the
price-taker's output, a price-taker can triumph over her rivals in terms of
major performance measures utilized in economics and business by adopting a
combination of price-taking strategy and growth-rate adjustment strategy.
Moreover, it is also the price-taker who is in control of the complexity of the
oligopolistic dynamics. Numerical simulations successfully verified all
theoretical conclusions.In our analysis, the Cournot optimizers are assumed to command the full knowledge of the
price-taker's current production. Such assumption
reduces the oligopolistic dynamical system into a simple one-dimensional
discrete process and hence greatly simplifies the analysis. If the Cournot
optimizers know only the price-taker's outputs in previous periods, then a
multidimensional discrete process is formed. Global analysis as implemented in [10] should be carried out.Apparently, the current studies can be generalized in
many ways. For instance, it is interesting to see the relative profitability of
the different agents in an oligopolist economy with product differentiation.
Analogous analysis can also be conducted for heterogeneous oligopsonist model.
However, the analysis turns to be much complicated
when multiple nontrivial equilibria may appear, under which a suitable “selection mechanism” as discussed in [11] needs to be
considered.



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*Source: 905143-2008-09-29.xml*