Similarly, for any specified value of A’s output, the corresponding value of B’s output maximises B’s profit. KJQN202000832), High-Level Talents Program of Chongqing Technology and Business University (Grant No. It thus sells ½OB. In addition to the important impact of parameters changes on equilibrium output with different adjustment mechanisms, we should also consider the impact of the previous output, especially the production quantity in the last phase. Both rms have the same unit production cost c = 30. A. Elsadany and A. M. Awad, “Dynamical analysis and chaos control in a heterogeneous kopel duopoly game,”, H. N. Agiza and A. He criticised and rejected both the Cournot and Bertrand cases on the ground that none of them conforms perfectly to the hypothesis that each seller acts so as to render his profit a maximum. The Cournot solution is unrealistic because it assumes zero cost of production. Stackelberg Model of Duopoly Stackelberg’s Model of Duopoly also has to do with companies trying to decide how much of a homogeneous good to produce. cstc2019jcyj-msxmX0470), Science and Technology Research Project of Chongqing Municipal Education Commission (Grant No. In static Cournot oligopoly games, all firms know other opponents’ strategy space, payoff functions, and make actions simultaneously, which means each firm adopts a naïve expectation to make his production decision, so he assumes that the opponents’ yield keeps the same level as that in the previous period. Thus there is no equilibrium situation. R&D spillovers, R&D investments, and technological innovation efficiency are introduced in our model. To verify our theoretical results, numerical simulations are carried on to show complex dynamic in Section 4, such as maximum Lyapunov exponents, bifurcations, strange attractors, and sensitive dependences on initial conditions. Model ini dikembangkan oleh ekonom Jerman Heinrich von Stackelberg dan merupakan perpanjangan dari model Cournot. (10) At the same time, each seller takes the supply or output of its rival as constant. A. Hołyst and K. Urbanowicz, “Chaos control in economical model by time-delayed feedback method,”, Y. The Stackelberg model includes the following four possibilities. Thus the reaction curves of the Stackelberg solution, based as they are on conjectures, have made his theory weak and unrealistic. Share Your Word File In this rival’s price increases. Various methods for controlling chaos have been used in dynamical systems; the OGY method was presented by Ott et al. Industrial Organization-Matilde Machado Stackelberg Model 11 3.3. We are committed to sharing findings related to COVID-19 as quickly as possible. Part I,”, T. Li and J. Ma, “The complex dynamics of R & D competition models of three oligarchs with heterogeneous players,”, H. Tu and X. Wang, “Complex dynamics and control of a dynamic R&D Bertrand triopoly game model with bounded rational rule,”, J. Zhou, W. Zhou, T. Chu, Y.-x. When В raises his price from LK upward, A follows him by raising his price from LB upward, till both reach the equilibrium point E. Thus the solution is again determinate. Firm A finds that with the entry of B, price has fallen to OP1 from OP. 3. In this paper, the duopoly Stackelberg–Cournot game is divided into two stages. Basic properties of the discrete dynamical system have been analyzed numerically via computing Lyapunov exponents, bifurcation diagrams, sensitive dependence on initial conditions, strange attractors, and chaos controlling. Once the price OP1, is set, no one will cut it, for each seller realizes that by so doing, he will reduce his profit. These papers, which studied on Stackelberg–Cournot games or Cournot–Stackelberg games, are primarily based on perfect rationality. In Section 2, the nonlinear duopoly Stackelberg–Cournot model is described, and a two-dimensional discrete system with heterogeneous players is formulated. Complexity behaviors, such as period doubling and chaotic attractors, will also occur, when the maximum Lyapunov exponents of the system (13) are positive. (8) Each seller decides about the quantity he wants to produce and sell in each period. A. Amer, “Resonance and vibration control of two-degree-of-freedom nonlinear electromechanical system with harmonic excitation,”, X. S. Luo, G. Chen, B. Hong Wang, and J. Qing Fang, “Hybrid control of period-doubling bifurcation and chaos in discrete nonlinear dynamical systems,”. That means, on one hand, R&D spillovers can lower the enthusiasm of firms because of other firms’ “hitchhike”, and on the other hand, it could also reduce all firms’ production cost due to the positive externalities. From Figure 6, we can see that the time series of system (13) is sensitive dependent on initial conditions, i.e., complex dynamics behaviors occur in this model. Finally, our research gives the relationship between Nash equilibrium output and R&D input in a completely rational monopoly market and provides the region where the equilibrium output exists in a boundedly rational duopoly. Figure 5 shows the graph of a strange attractor of the dynamic game (13) for the parameter values . (9) But each is ignorant about his rival’s plan about output. It, therefore, regards SD1 segment of the market demand curve as its demand curve. Therefore, in order to facilitate a stable output, the R&D leader can slow its output adjustment speed, with other parameters, as shown in Figure 1; they can also enhance the atmosphere of technology sharing, as shown in Figure 2; besides, the R&D follower can improve its innovation efficiency, as shown in Figure 4. Firm 1 is the Stackelberg leader, and firm 2 is the follower; besides, we denote the output of firm at time period by , which is updated according to discrete time steps. But the price under monopoly (OP) would be higher than under the Cournot solution (OP2). The Chamberlin solution involves a kind of agreement between the two sellers. (7) Both have identical costs and identical demands. Many works focus on games with homogeneous strategies [3, 4, 16], but the belief that each firm in oligopolistic market behaves with different expectation [6–8, 10, 13, 25, 26], is common in real word, and as a consequence, our paper will apply this belief to a duopoly game. Stackelberg’s Duopoly 5. Thus, Bertrand argued that there would not be any limit to the fall in price since each seller could by doubling his produce, and underbid his rival. Hang, and H. Yang, “Analysis of the dynamics of multi-team Bertrand game with heterogeneous players,”, X. Pu and J. Ma, “Complex dynamics and chaos control in nonlinear four-oligopolist game with different expectations,”, C. D’Aspremont and A. Jacquemin, “Cooperative and noncooperative R & D in duopoly with spillovers,”, A. Tesoriere, “Endogenous R & D symmetry in linear duopoly with one-way spillovers,”, G. I. Bischi and F. Lamantia, “A dynamic model of oligopoly with R&D externalities along networks. No doubt the leadership equilibria (cases 1 and 2) contain an element of collusion or of spontaneous coordination, yet they represent an arbitrary form of coordination when leadership expresses itself in selecting a point along a reaction curve of the traditional kind. A Model of Duopoly with Stackelberg Equilibrium By Takashi Negishi and Koji Okuguchi, Tokyo and Yokohama, Japan (Received February 14, 1972) Stackelberg disequilibrium for duopoly disappears if the assump-tion of the perfect information is dropped and each firm is assumed to estimate the reaction function of the rival which will be shifted as Problem 4 (24 marks) Consider a homogenous-product Cournot duopoly model in which Q is the market output, 1 and 2 Q are the output levels produced by firms 1 and 2, respectively, and P is the resulting uniform market price. We set in (13), and we can get the solution of the nonlinear algebraic system as follows: Easily, the algebraic system (14) has two equilibrium points:where the expressions of and are given by equations (8) and (9), respectively and and are shown in equation (10). [15], hold:The abovementioned inequalities of , , and define a region where the Nash equilibrium point is locally stable. If they enter into a collusive agreement, they may maximise their profits jointly. The following points highlight the top three models of duopoly. We examine a desirable role (either leader or follower) of the public firm. The Cournot and Stackelberg duopoly theories in managerial economics focus on firms competing through the quantity of output they produce. A Dynamic Stackelberg–Cournot Duopoly Model with Heterogeneous Strategies through One-Way Spillovers, School of Management Science and Engineering, Chongqing Technology and Business University, Chongqing 400067, China, School of Business, Hunan Agricultural University, Changsha 410128, Hunan, China, S. S. Askar and T. Simos, “Tripoly stackelberg game model: one leader versus two followers,”, W. J. Baumol and R. E. Quandt, “Rules of thumb and optimally imperfect decisions,”, G. I. L. F. Bischi, “Nonlinear duopoly games with positive cost externalities due to spillover effects,”, E. M. Elabbasy, H. N. Agiza, and A. 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. But problems might arise in sharing profit by the two sellers. We also consider endogenous roles by adopting the observable delay … Each act rationally, looks beyond his nose and understands that sharing monopoly profit is to the best of his advantage. In Section 5, a new method named control strategy of the state variables feedback and parameter variation is employed to control chaos of the system. It may result from the assumption that the rival seller moves along a reaction curve which, in reality, does not exist for him. The models are: 1. Two types of heterogeneous players, who adopt the bounded rational expectation and naïve mechanism, respectively, determine their R&D investments sequentially in the Stackelberg R&D phase and make output decisions simultaneously in the Cournot production phase. Figure 7 indicates that system (13) can get rid of chaos successfully when the controlling parameter reaches 0.298, and Figure 8 shows that the chaotic system is controlled at a fixed point when . Copyright © 2020 Jianjun Long and Hui Huang. The underestimation of the market demand curve leads to the wrong estimation of the market MR curve. These reaction functions are derived from the profit indifference maps of the individual duopolists. What is the Stackelberg Model? Kinked Demand Curve Diagram. The content of this paper is as follows. The leader uses bounded rationality, he does not have a complete knowledge of the market demand function, and determines his output on the basis of expected marginal profit ; in consequence, he increases (decreases) the production if the marginal profit is positive (negative) at the next period. In the chaotic states, the market will sensitively depend on the parameters values, and parameter variations will lead to the markets’ long-term trajectory unpredictable. [40] and had been applied in the dynamic game model to control chaos [41, 42], a modified straight-line stabilization method [12], adaptive control [13], time-delayed feedback method [43], and other feedback control methods [6–8] had also been studied for the chaos control in an economic model with homogeneous or heterogeneous expectations. But, as a matter of fact, neither follows the reaction function of the other. Precisely because of the strong hypothesis that, the classical Cournot model and the typical Stackelberg model require perfect rationality, is difficult to achieve in reality [4], expectations turn into an important role in modeling economic phenomena, and each oligopolistic producer can use different expectation to adjust his output. In the Stackelberg duopoly model, one firm determines its profit-maximizing quantity and other firms then react to that quantity. Expected profits decline to OP JA functions are derived from the other suppose there are mineral! Satisfied by the two sellers duopoly game with one-way spillovers ( OP2 ) General. For controlling chaos have been used in dynamical systems stackelberg duopoly model diagram the OGY method presented. With Chamberlin that monopoly solution is possible under duopoly interdependence, looks beyond his nose and that! Becomeswe can obtain the threshold given by of competition on output with stochastic perturbations is proposed of! 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stackelberg duopoly model diagram

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