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Discrete Dynamics in Nature and Society
Volume 2008, Article ID 905143, 20 pages
http://dx.doi.org/10.1155/2008/905143
Research Article

Triumph over Your Rivals in Dynamic Oligopoly

Division of Economics, School of Humanities and Social Science, Nanyang Technological University, Nanyang Avenue, Singapore 639798

Received 23 March 2008; Accepted 10 July 2008

Academic Editor: Akio Matsumoto

Copyright © 2008 Weihong Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 Section 2 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 which 𝑁=1+𝑚 firms produce a homogeneous product with quantity 𝑞𝑖𝑡,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=𝑁𝑖=1𝑞𝑖𝑡, 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 function 𝐶, with 𝐶>0 and 𝐶>0, they adopt different production strategies. We assume that all firms can be classified into two categories: the price-takers and the Counortoptimizers.

The first firm is the price-taker, who is either deficient in market information or less strategic in market competition. Its production target, denoted by ̂𝑥𝑡 is derived by equating the marginal cost (i.e., the first-order derivative of cost) with market price if last period, that is, 𝐶(̂𝑥𝑡)=𝑝𝑡1. Notice that 𝑝𝑡1=𝐷(𝑄𝑡1), the production target ̂𝑥𝑡 can be expressed aŝ𝑥𝑡=𝑀𝐶1𝐷𝑄𝑡1,(2.1)where 𝑀𝐶1 denotes the inverse function of 𝐶.

Instead of producing at ̂𝑥𝑡, 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 from𝑥𝑡𝑥𝑡1𝑥𝑡1=𝛾̂𝑥𝑡𝑥𝑡1,(2.2)or, equivalently, from𝑥𝑡=𝑥𝑡11+𝛾𝑀𝐶1𝐷𝑄𝑡1𝑥𝑡1.(2.3)

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 rest 𝑚 firms are Counort optimizers. 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 by𝜋𝑦𝑥𝑡,𝑦𝑡=𝑠𝑦𝑥𝑡,𝑦𝑡𝑦𝐶𝑡𝑄=𝐷𝑡𝑦𝑡𝑦𝐶𝑡,(2.4)where 𝑠𝑦(𝑥𝑡,𝑦𝑡)=𝑝𝑡𝑦𝑡=𝐷(𝑄𝑡)𝑦𝑡 is the sales revenue.

Then the first-order profit maximization condition 𝜕𝜋𝑦/𝜕𝑦𝑡=0 yields𝐷𝑄𝑡+𝑄𝜕𝐷𝑡𝜕𝑦𝑡𝑦𝑡𝐶𝑦𝑡=0.(2.5)

Since 𝑄𝑡=𝑥𝑡+𝑚𝑦𝑡, we have 𝜕𝐷(𝑄𝑡)/𝜕𝑦𝑡=𝑚𝐷(𝑄𝑡) and so that (2.5) is simplified to 𝐷𝑄𝑡+𝑚𝑦𝑡𝐷𝑄𝑡=𝐶𝑦𝑡.(2.6)

Assume that appropriate second-order condition is satisfied so that 𝑦𝑡=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 general heterogeneous oligopoly model is said to be economically meaningful, if the following inequalities are met:
(i)𝑥𝑡0 and 𝑦𝑡0, 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 a linear heterogeneous oligopoly model (LHO model), in which the market demand is assumed to be linear so that its inverse demand function is given by𝑝𝑡𝑥=𝐷𝑡+𝑚𝑦𝑡=1𝑥𝑡𝑚𝑦𝑡,(3.1)whereas the marginal cost is linear so that the cost function adopts the form of𝐶(𝑞)=𝐶0+𝑐2𝑞,(3.2)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) yields𝑥𝑡=𝑥𝑡11+𝛾1𝑥𝑡1𝑚𝑦𝑡1𝑐𝑥𝑡1.(3.3)

On the other hand, (2.6) defines an optimal reaction function for the Cournot optimizers:𝑦𝑡=𝑅𝑦𝑥𝑡=1𝑥𝑡2𝑚+𝑐.(3.4)

Substituting (3.4) into (3.3) and manipulating them yield a quadratic recurrence relation𝑥𝑡𝑥=𝑓𝑡1𝛾𝑥̇=𝑎𝑡1𝛾𝑥1𝑏𝑡1,(3.5)where𝑎𝛾𝛾̇=1+𝑐+𝑚𝑐,𝑏𝛾𝑐+2𝑚̇=𝛾𝑐+𝑚+𝑐𝑐+2𝑚.𝑐+𝑚𝛾+𝑐𝑐+2𝑚(3.6)

For the discrete process (3.5), the trajectory {𝑥𝑡}𝑡=0 is bounded by 𝑥0(𝛾)=1/𝑏(𝛾) if and only if 0𝑥0𝑥0(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, if1<𝛾𝛾max,(3.7)where𝛾max=3𝑐𝑐+2𝑚𝑐+𝑚.(3.8)

Figure 1 illustrates the impact of 𝛾 on the functional shape of 𝑓 for the case of 𝑐=1. We see that𝜕𝑥0𝛾𝑐𝜕𝛾=𝛾2𝑐+2𝑚𝑐+𝑚+𝑐𝑐+2𝑚<0,𝜕𝑥0𝛾=𝑐𝜕𝑚2𝛾1𝛾𝑐+𝑚+2𝑐𝑚+𝑐22<0if𝛾>1.(3.9)

fig1
Figure 1: Illustration of 𝑥𝑡=𝑓(𝑥𝑡1), relative profitability regime Ω𝑝 and the trapping set Ω(𝛾).

The 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 equilibrium 𝑥:𝑥=𝑚+𝑐𝑚+𝑐+2𝑚𝑐+𝑐2,(3.10)at which the derivative of 𝑓 is given by𝑥𝜎̇=𝑑𝑓|||𝑑𝑥𝑥𝛾=1𝑐+𝑚𝑐𝑐+2𝑚;(3.11)(ii)the equilibrium 𝑥 is stable for 𝛾<𝛾1, and the converge to the equilibrium is monotonic if 𝛾<𝛾0 and cyclical if 𝛾>𝛾0, where𝛾0=𝑐𝑐+2𝑚𝛾𝑐+𝑚,(3.12)1=2𝛾0;(3.13)𝛾(iii)increasing 𝛾1 from 𝛾 to 𝛾2.5699𝛾0(3.14), where2𝑛will lead to a sequence of period-doubling bifurcations, that is, stable cycles with periods 2𝑛1 appear right after the cycles with periods 𝑛1 become unstable, for 𝛾;(iv)increasing 𝛾 from 𝛾max to 𝛾, stable cycles for all periods can appear at some specific intervals for 𝛾 while aperiodic oscillations occur for the rest value of 𝛾=𝛾max;(v)for 𝜑, 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 density 𝜑𝑓𝑥=1𝜋𝑥𝑥max𝑥for𝑥𝐽̇=[0,𝑥max],(3.15)that is absolutely continuous with respect to the Lebesgue measure, where𝑥max=𝑥0𝛾max=4𝑐+𝑚3𝑐+𝑚+2𝑐𝑚+𝑐2;(3.16)where𝛾>𝛾max(vi)for 𝑥0>0, the trajectory starting from almost all 𝑓 becomes unbounded and hence is not economically meaningful.
Proof. At first, we point out that the map 𝛾=𝛾max,𝑓 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) When 𝑋𝑡𝑥=𝐹𝑡1̇=4𝑋𝑡1(1𝑋𝑡1)(3.17) is topologically conjugated with𝑓=1𝐹in the sense that 𝐽[0,1], where 𝑥=𝑥𝑥max,for𝑥𝐽.(3.18) is a homeomorphism defined by𝜑𝐹(𝑋)
By the law of conjugation, if 𝐹 is the invariant density for 𝑓, then the invariant density for 𝜑𝑓(𝑥)=𝜑𝐹|||(𝑥)𝑑|||𝑑𝑥.(3.19) is given by𝜑𝐹1(𝑋)=𝜋𝑋1𝑋for𝑋[0,1].(3.20)
It is well known that the invariant density preserved by (3.17) is given by
Substituting with [𝛾,𝛾max] 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 𝑥 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-state  𝛾 is independent of the adjustment rate 𝛾, the stability does depend on 𝛾>𝛾0. For a fixed 𝑥, the stability of 𝜕𝜎𝜕𝑐=𝛾𝑐+𝑚2+𝑚2𝑐2𝑐+2𝑚2>0,𝜕𝜎=𝛾𝜕𝑚𝑐+2𝑚2>0.(3.21) is characterized by the derivative given by (3.11), from which we have𝑐Thus, increasing number of Cournot optimizers stabilizes the economy, but increasing the technological level (less 𝛾) destabilizes the economy.

Figures 2(a) and 2(b) depict the bifurcation diagram with respect to 𝑐 and 𝑥𝑡=𝑓(𝑥𝑡1),𝑚=3., respectively. A structural similarity is apparently observed between them.

fig2
Figure 2: Bifurcations of 𝛾(𝑐=1/2)

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(𝑚+𝑐)/(2𝑚+𝑐)<1that 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 𝜋𝑥𝑡=𝑝𝑡𝑐𝑥𝑡2𝑥𝑡=𝑚+𝑐22𝑚+𝑐1𝑚+𝑐+𝑐𝑐+2𝑚2𝑥𝑚+𝑐𝑡𝑥𝑡,𝜋𝑦𝑡=𝑝𝑡𝑐𝑦𝑡2𝑦𝑡=𝑐𝑚+2𝑦2𝑡=1𝑥𝑡22,2𝑚+𝑐(4.2)).

The profits made by the price-taker and by any of the Cournot optimizers are𝜋𝑦𝑡0respectively.

We see that, 𝑡 for all 𝜋𝑥𝑡>0, 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, 𝑥𝑡<𝑥+ if and only if 𝑥+, where 𝑥+=2𝑚+𝑐2𝑚+𝑐+𝑐2𝑚+𝑐<1.(4.3) is defined byΔ𝜋𝑡𝑥𝑦̇=𝜋𝑥𝑡𝜋𝑦𝑡=𝑐+2𝑚+11+𝑐2𝑥2𝑚+𝑐𝑥𝑡𝑥𝑡𝑥,(4.4)

Simple mathematical manipulations indicate that𝑥̇=1/(1+𝑐+2𝑚)where 𝑥̇=1/(1+𝑐) and 𝜋𝑥𝑡𝜋𝑦𝑡.

Therefore, 𝑥𝑡Ω𝑝̇=[𝑥,𝑥].Ω𝑝 if and only if 𝛾 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, 𝑠𝑥𝑡=𝑝𝑡𝑥𝑡=𝑚+𝑐𝑥2𝑚+𝑐𝑡1𝑥𝑡,𝑠𝑦𝑡=𝑝𝑡𝑦𝑡=𝑚+𝑐2𝑚+𝑐21𝑥𝑡2.(4.5), 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 by𝑠𝑥𝑡

We see that 𝑠𝑦𝑡 and 𝑥𝑡<1 are always positive (due to the fact that Δ𝑠𝑡𝑥𝑦̇=𝑠𝑥𝑡𝑠𝑦𝑡=𝑝𝑡𝑥𝑡𝑦𝑡,(4.6) is implicitly guaranteed). Moreover, sinceΔ𝑠𝑡𝑥𝑦>0 we have 𝑥𝑡>𝑦𝑡 so long as 𝑥𝑡>𝑥=11+𝑐+2𝑚,(4.7), which occurs when Ω𝑝that is, when the price-taker's output exceeds the lower bound of the relative profitability regime 𝑥.

4.3. Relative Performance

At the unique nontrivial equilibrium 𝑥Ω𝑥 given by (3.10), the other relevant equilibrium values are summarized in the second column of Table 1. It can be verified that 𝑐, regardless of the values of 𝑚, 𝛾, and 𝑝𝑤=𝑐1+𝑚+𝑐. Therefore, at the equilibrium, the price-taker enjoys a higher sale revenue as well as a higher profit than the Cournot optimizers.

tab1
Table 1: Equilibrium outcomes.

For benchmarks, two extreme situations are also included in Table 1, one with all 𝑚+1 firms adopting the price-taking strategy and the other with all 𝑞𝑢=12𝑚+2+𝑐,(4.8) firms forming a collusion.

In the former case where all firms behave as the price-taker, the equilibrium is known as a Walrasian equilibrium (competitive equilibrium). For the latter case, all firms behave as if they were a monopolist and maximize the total profit with a constant output𝑚=1from which the other performance measure can be evaluated. (Again we do not consider the case in which the monopoly maximizes the sales revenue.)

While Figure 3 provides graphical illustrations of the outputs, the sales revenue, and the profits at the equilibrium with 𝑚=3 and 𝑚=1.

fig3
Figure 3: Critical outputs and performance measures.

In 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 by𝑄𝑡̇=𝑥𝑡+𝑚𝑦𝑡𝐶>0where 𝐶>0, the cost function is strictly convex ((𝐷<0) and 𝐷(𝑄𝑡)) and the demand function 𝐷(𝑄𝑡) satisfies the additional assumptions that both 𝑄𝑡0 and 𝑥>𝑦>0 are bounded for 𝑠𝑥>𝑠𝑦>0, the following facts hold.
(i)All nontrivial intertemporal equilibria are economically meaningful.(ii)𝜋𝑥>𝜋𝑦 so that 𝜋𝑥>𝜋𝑦>0.(iii)𝐶(0)=0. Moreover, (𝑥,𝑦) if 𝑥>0.
Proof. (i)-(ii) At a nontrivial intertemporal equilibrium (steady-states 𝑝=𝐶(𝑥).(4.11)), we mean 𝐶>0. It follows from (4.9) that we must have𝑝>0
The facts that 𝑦 thus implies 𝐶(𝑥)+𝑚𝐷𝑄𝑦=𝐶𝑦,(4.12). What remains to be verified is the positiveness of 𝑄=𝑥+𝑚𝑦. Equation (4.10) yields𝐷(𝑄)where 𝑦=0. Since 𝑦=0 is bounded even when 𝐶(𝑥)=𝐶𝑦=𝐶0.(4.13), a hypothesis that 𝑥=0 would lead to𝐶However, (4.13) in turn suggests that 𝑥>0 by the monotonicity characteristics implied from the convexity assumption of 𝑦, which is a contradiction to the fact that 𝐷<0.
Again from (4.12), a positive 𝑥>𝑦 and 𝑝, together thus suggest 𝑠𝑥>𝑠𝑦>0. Under the same equilibrium price Δ𝜋𝑥𝑦=𝜋𝑥𝜋𝑦=𝐶(𝑥)(𝑥𝑦)𝐶(𝑥)𝐶(𝑦).(4.14) we thus have 𝐶()>0.
(iii) For the profit difference at the nontrivial equilibrium, (4.11) suggests that𝑥𝑦
It follows from the assumption of 𝐶(𝑥)(𝑥𝑦)𝐶(𝑥)𝐶(𝑦)>0 and 𝜋𝑥>𝜋𝑦.(4.15) that 𝜋𝑦>0, or equivalently,𝑝=𝐶(𝑦)𝑚𝐷(𝑄)𝑦
Next we need to show that 𝜋𝑦=𝑝𝑦𝐶(𝐶𝑦)=𝑦𝑚𝐷𝑄𝑦𝑦𝐶(𝑦)=𝐶𝑦𝑦𝐶(𝑦)+𝑚𝐷𝑄𝑦2,(4.16) if the fixed cost is negligible. It follows from (4.12) that 𝐶(𝑦)𝑦>𝐶(𝑦) so that𝐶(0)=0which is positive since 𝑦𝑘,𝑘=1,2,,𝑚 if 𝑥.
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, 𝛾, is different from 𝛾>𝛾. 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. When 𝛾>𝛾1 is sufficiently large (𝐺𝐶1 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 Section 3, when 𝑥𝑡, 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. Let 𝛾 be a performance measure function of 𝑘. Given an adjustment speed {𝑥𝑖(𝛾)}𝑘𝑖=1 and an associated periodic-𝐺𝑘𝛾1̇=𝑘𝑘𝑖=1𝐺𝑥𝑖𝛾,(5.1) cycle {𝑥𝑡}𝑡=0, its k-period average is defined as𝐺𝛾̇=lim𝑇1𝑇𝑇𝑡=0𝐺𝑥𝑡.(5.2)and its long-run average for a trajectory {𝑥𝑖(𝛾)}𝑘𝑖=1 is defined as𝐺𝛾=𝐺𝑘𝛾.(5.3)

For the discrete dynamic process (3.5), if 𝑘 is a stable periodic cycle, then we have𝛾(0,𝛾max)

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-𝛾max 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 model given by (3.5) and (3.4), for 𝐺𝑥𝑘𝛾𝐺>0,𝑦𝑘𝛾>0forany𝑘1,(5.4), where 𝐺𝑥𝛾𝐺>0,𝑦𝛾>0,(5.5) is defined in (3.8), one always has𝐺and more generally, 𝜋where 𝑠 denotes 𝑠 and 𝑥𝑡>0.Proof. The conclusion for the sales revenue 𝑡 follows straightforwardly from the fact that 𝛾(0,𝛾max) for all 𝑥0<𝑥0(𝛾) if 𝜋 and 1𝛾𝑥𝑡+1𝑥𝑡1+𝑥𝑡=𝑀𝐶1𝑝𝑡,(5.6), as discussed in Section 3.
For the profit 𝑝𝑡=𝑐𝛾𝑥𝑡+1𝑥𝑡1+𝑐𝑥𝑡,(5.7), we note first that (4.9) implies that𝜋𝑥𝑥𝑡=𝑝𝑡𝑥𝑡𝑥𝐶𝑡=𝑐𝛾𝑥𝑡+1𝑥𝑡+𝑐2𝑥2𝑡.(5.8)that is,{𝑥𝑖}𝑘𝑖=1which leads to𝜋𝑥𝑘𝛾1̇=𝑘𝑘𝑖=1𝜋𝑥𝑥𝑖=1𝑘𝑘𝑖=1(𝑐/𝛾)𝑥𝑖+1𝑥𝑖+(𝑐/2)𝑥2𝑖=𝑐𝑘𝛾𝑥𝑘+1𝑥1+𝑐2𝑘𝑘𝑖=1𝑥2𝑖=𝑐2𝑘𝑘𝑖=1𝑥2𝑖>0,(5.9)
For a given periodic-k cycle 𝑥𝑘+1=𝑥1, we have{𝑥𝑡}𝑡=1where we utilize the fact that 𝜋𝑥𝛾̇=lim𝑇1𝑇𝑇𝑡=1𝜋𝑥𝑥𝑡=𝑐𝛾lim𝑇1𝑇𝑥𝑇𝑥1+𝑐2lim𝑇1𝑇𝑇𝑖=1𝑥2𝑖=𝑐2lim𝑇1𝑇𝑇𝑖=1𝑥2𝑖>0,(5.10).
Similarly, for any chaotical trajectory lim𝑇(1/𝑇)(𝑥𝑇𝑥1), the long-run average of profit for the price-taker can be recast as𝑥𝑇which holds because the limit term 𝑥1 approaches zero since both Ω𝑝 and 𝛾 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 Section 4, we see that there exists a relative profitability regime 𝑚 that is independent of the adjustment rate 𝑐 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 Figure 1, with fixed 𝛾0<𝛾𝛾max and 𝛾0, for 𝛾max, where Ω(𝛾)̇=[𝑥min(𝛾),𝑥max(𝛾)] and 𝑥0(0,𝑥0(𝛾)) are, respectively, defined in (3.12) and (3.8), there exists a trapping set {𝑥𝑡}𝑡=1 such that starting from any initial output Ω(𝛾), the output trajectories 𝑥min(𝛾) will be eventually trapped in 𝑥max(𝛾) and remain there forever. 𝑥max𝛾𝑥=𝑓0𝛾2=𝛾𝑐+𝑚+𝑐𝑐+2𝑚2,𝑥4𝑐𝛾𝑐+2𝑚𝑐+𝑚+𝑐𝑐+2𝑚min𝛾𝑥=𝑓max𝛾=𝛾𝑐+𝑚+𝑐𝑐+2𝑚33𝑐𝑐+2𝑚𝛾𝑐+𝑚16𝑐3𝛾𝑐+2𝑚3.𝑐+𝑚+𝑐𝑐+2𝑚(5.11) and 𝛾<𝛾max,𝑥min(𝛾)>0. are determined from𝜕𝑥max(𝛾)/𝜕𝛾>0

For 𝜕𝑥min(𝛾)/𝜕𝛾<0 Moreover, 𝛾0𝛾𝛾max and 𝑥min(𝛾0)=𝑥max(𝛾0)=𝑥 for Ω(𝛾).

From the fact that 𝑥, we can infer that the trapping set 𝛾 grows around 𝜖 along with increasing value of 𝛾=𝛾0+𝜖. This in turn suggests that for a sufficiently small positive Ω(𝛾)Ω𝑝 and 𝛾, we are able to enforce 𝑠𝑥𝑡>𝑠𝑦𝑡. That is to say, by limiting the magnitude of 𝑥𝑡>𝑥, 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 for Ω(𝛾) is just Ω𝑝, which can be guaranteed as long as the lower-bound of 𝑥min𝛾𝑠=𝑥(5.12) coincides with the lower-bound of 𝛾𝑠, solving the identity𝑠𝑥𝑡>𝑠𝑦𝑡yields an upper bound 𝑥𝑡Ω(𝛾) for the adjustment rate to ensure 𝛾 for all 𝛾.

Denote by  Ω(𝛾) the maximum possible Ω𝑝 such that the upper bound of the trapping set 𝛾 coincides with the upper bound of the relative profitability regime 𝑥max(𝛾)=𝑥, then 𝛾𝑝̇=min{𝛾𝑠,𝛾} is a solution to the identity 𝛾𝑠𝛾𝑝.

Finally, let 𝑚, then we have 𝑐 by definition and the following beautiful results.

Theorem 5.3. With the LHO model defined in (3.4) and (3.5), for any given 𝛾[𝛾0,𝛾𝑝] and 𝜋𝑥𝑡>𝜋𝑦𝑡>0,
(i)when 𝑥𝑡Ω(𝛾), one has  𝛾[𝛾0,𝛾𝑠] for all 𝑠𝑥𝑡>𝑠𝑦𝑡>0, 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)when 𝑥𝑡Ω(𝛾), one has  𝛾𝑠 for all 𝛾𝑝, 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.

Table 2 provides a list of 𝛾𝑝 and 𝛾𝑠 values for several typical sets of parameters we have applied in our illustrations.

tab2
Table 2: Some typical values.

When 𝛾𝑠 exceeds the upper bounds Δ𝜋𝑡𝑥𝑦 and Δ𝑆𝑡𝑥𝑦, respectively, Δ𝜋𝑥𝑦𝛾>0 and 𝑥𝑡Ω(𝛾) 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 Theorem 5.3 that 𝛾[𝛾0,𝛾𝑝] is guaranteed for all 𝜋𝑥𝑡, if 𝜋𝑦𝑡. By the continuity of 𝛾 and 𝛾𝑝𝑘 as functions of 𝛾0𝛾<𝛾𝑝𝑘 and the ergodicity of the dynamical process (3.5), we can hypothesize that there exists a Δ𝜋𝑡𝑥𝑦<0 such that so long as 𝑡, despite that Δ𝜋𝑘𝑥𝑦𝛾>0 for some 𝛾𝑝𝑘, we still have 𝑘 for any cyclic orbit. Apparently, the value of 𝑥1(𝛾),𝑥2(𝛾) depends on the order 𝑥1(𝛾)𝑥2(𝛾). For instance, for a period-2 cycle (Δ𝜋2𝑥𝑦𝛾=12Δ𝜋𝑥𝑦𝑥1𝛾+Δ𝜋𝑥𝑦𝑥2𝛾122𝑖=1𝑥𝑥𝑖𝛾𝑥𝑖𝛾𝑥.(5.13)), with 𝛾, (4.4) suggests that𝑥1(𝛾0)=𝑥2(𝛾0)=𝑥The RHS of (5.13) is a continuous function of 𝛾0. In considering the fact that 𝜕𝑥1(𝛾)/𝜕𝛾>0, where 𝜕𝑥2(𝛾)/𝜕𝛾<0 is defined in (3.12), and the differential properties 𝛾1𝛾𝛾max and 𝛾𝑝2 for 𝛾𝑝2>𝛾0, we are able to conclude the existence of a Δ𝜋𝑥𝑦(𝛾)2=0, 𝛾>𝛾, such that the equality 𝛾𝑝 holds.

Similar conclusion can be expected for an aperiodic orbit which occurs when 𝛾<𝛾𝑝. There exists a Δ𝜋𝑥𝑦𝛾>0 value such that for all 𝛾𝑝, we have 𝑚. There is no analytical formula to obtain the value of 𝑐.

Figures 4 to 6 depict the numerical simulations of relevant performance measures for several combinations of 𝛾(𝑚=3,𝑐=1/2) and 𝛾𝑝 values.

fig4
Figure 4: Long-run averages with respect to 𝑐.

Comparing Figure 4(a) with Figure 5(a), we see that 𝑐 decreases with the increase in 𝛾𝑝. For a relatively small 𝛾max (a more divergent economy caused by higher marginal cost), 𝑚 can even exceed its maximum possible growth-rate limit 𝛾𝑝, as illustrated in Figure 4. On the other hand, by comparing Figure 4(a) with Figure 6(a), we see that increasing 𝛾max (more Cournot optimizers) stabilizes the economy so that 𝛾(𝑚=3,𝑐=1) increases along with the increase of the maximum possible growth-rate limit 𝛾(𝑚=1,𝑐=1/2).

fig5
Figure 5: Long-run averages with respect to 𝑠𝑥𝛾>𝑠𝑦𝛾.
fig6
Figure 6: Long-run averages with respect to 𝛾<𝛾max.

Similarly, there exist cases in which 𝑚 for all 𝑐, 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 𝛾𝑝,𝛾𝑝>𝛾𝑝 and 𝛾[𝛾0,𝛾𝑝],
(i)there exists Δ𝜋𝑥𝑦𝛾>0, such that for all 𝛾𝑠,𝛾𝑠𝛾𝑠, one has  𝛾[𝛾0,𝛾𝑠], that is, the long-run average profits made by the price-taker is higher than that earned by each individual Cournot optimizer;(ii)there exists Δ𝑠𝑥𝑦𝛾>0, such that for all 𝛾=𝛾max, one has  𝜑𝑓, that is, the long-run average sales-revenue earned by the price-taker is higher than that earned by each individual Cournot optimizer.

When 𝐺𝐶1, process (3.5) becomes full-chaotic and preserves an ergodic invariant density 𝐺𝛾=𝑥max0𝐺𝑥𝜑𝑓𝑥𝑑𝑥.(5.14) given by (3.15) so that for any function Δ𝜋𝑡𝑥𝑦, its space mean equals to the time mean in the sense ofΔ𝜋𝑥𝑦𝛾max=𝑥max0Δ𝜋𝑥𝑦𝑥𝜑𝑓𝑥𝑑𝑥𝑥max0𝑥𝑥𝑥𝑥𝜑𝑓𝑥𝑑𝑥𝑥max0𝑥𝑥𝑥𝑥𝑥𝑥max𝑥𝑑𝑥.(5.15)

For instance, substituting 𝑥,𝑥 defined in (4.4) into (5.14) leads to𝑥max

As 𝑚, and 𝑐 all depend on 𝑐 and 𝑚=3, 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 𝛾, from which the impact of Δ𝜋𝑥𝑦𝛾max>0 is illustrated. Again, as expected, a range for 𝛾=𝛾max,𝑚=3 can be identified, in which we have .

fig7
Figure 7: Performance measures under full-chaos ().

Although our analysis is conducted using the LHO model, the methodology applied and conclusions drawn in Theorem 5.4 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.

Acknowledgments

This research is partly supported by Grant RG68/06 from Nanyang Technological University. The author is grateful for the editor in charge Akio Matsumoto for the efficient editorial work. The insightful suggestions from Laura Gardini improved the quality and the readability substantially. Appreciation also goes to Ong Qiyan for the technical assistances.

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