Exactly due to that, each oligopolistic enterprise must consider not only its own quantity decision but also the reactions of all other competitors, and behaviors of both Cournot games and Stackelberg games become more and more complicated. The German economist Stackelberg proposed a solution to the duopoly problem based on the assumption that each seller recognises the interdependence of other’s actions. The Nash equilibrium point is asymptotically stable if . This paper investigates a dynamic Stackelberg–Cournot duopoly game with one-way spillovers. Share Your Word File (4), e.g., the dynamical equation of firm 2 has the form as follows: We combine equations (11) and (12); therefore, the two-dimensional system that characterizes the dynamics of a Stackelberg–Cournot duopoly game with heterogeneous players is given by. As more and more firms enter the oligopoly industry, the equilibrium output and price of the industry will approach the perfectly competitive output OD1 and the zero prices. The dynamic adjustment mechanism can be modeled as follows:where is a positive constant, which represents the output adjustment speed of firm 1. Our research contributes to the extant literature on complex dynamics of Cournot or Stackelberg games. A. Yorke, “Erratum: “controlling chaos”,”, L. Kaas, “Stabilizing chaos in a dynamic macroeconomic model,”, H. N. Agiza, “On the analysis of stability, bifurcation, chaos and chaos control of kopel map,”, J. But the point selected on the reaction function by one seller does not play any part in shaping the policies of the rival seller. 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 .  discussed continuity properties of the followers reaction and provided sufficient conditions for existence of Stackelberg–Cournot equilibrium in oligopolistic markets. System (13) undergoes a flip bifurcation at when . 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. Joseph Bertrand, a French mathematician, criticising Cournot in 1883 pointed out that seller A in order to regain all the customers lost to B, will fix a price slightly below that fixed by В and price cutting may continue until the price becomes zero. This paper analyzes the dynamics of a Cournot duopoly model with different strategies. Se supone, por von Stackelberg, que un duopolista es lo suficientemente sofisticado como para reconocer que su competidor actúa bajo el supuesto de Cournot. 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,”. We also consider endogenous roles by adopting the observable delay … We consider a duopoly Stackelberg–Cournot game where two firms, labelled by , produce perfect substitute goods with production levels , respectively, and sell them at discrete time periods on a common market. Its corresponding marginal revenue curve is MR2 which intersects the horizontal axis (its marginal cost curve) at point B. Downloadable! Fellner does not agree with Chamberlin that monopoly solution is possible under duopoly interdependence. Then, condition (23) becomesWe can obtain the threshold given by the vanishing of the left-hand side of inequality (24). This research was supported by General Project of Chongqing Natural Science Foundation (Grant No. Mathematical properties of a stochastic Stackelberg–Nash–Cournot game  and a discontinuous Cournot–Stackelberg model  have been studied. Such leadership equilibria carry little meaning in relation to joint profit maximisation. Flåm et al. Our paper differs from these aforementioned references in three ways. To address these issues, our paper adopts a Stackelberg–Cournot model to analyze the decision-making process, which is divided into a Stackelberg R&D phase and a Cournot production phase. The Stackelberg Model 3. 1. Sign up here as a reviewer to help fast-track new submissions. In the kinked demand curve model, the firm maximises profits at Q1, P1 where MR=MC. Stage 1 is the Stackelberg R&D phase where the strategy space is the choice of R&D investments, two firms with different R&D capabilities sequentially carry out non-cooperative game around R&D investments for the sake of higher revenues, in the innovative process, the R&D leader makes decision on its R&D investments first, the follower determines his input after observing the opponent’s decision, and furthermore, we assume that R&D spillovers only flow from the R&D leader to the follower. We analyze two different scenarios: These papers, which studied on Stackelberg–Cournot games or Cournot–Stackelberg games, are primarily based on perfect rationality. This would tend to drive down the price to the competitive level in the long-run. is the coefficient of R&D spillovers, which implies that some benefits of firm 1’s R&D flow to firm 2 without payment, the external effect of the leader’s R&D is to lower the follower’s marginal production cost, specifically, and , means the technological innovation of the R&D leader is not freely obtained by the follower, while means fully obtained without any payment. Precisely, because the chaos in market are not expected and are even harmful to the participants, certain methods should be adopted to suppress or eliminate the occurrence of bifurcations and chaos. Diasumsikan, oleh von Stackelberg, bahwa satu duopolis cukup canggih untuk mengakui bahwa pesaingnya bertindak berdasarkan asumsi Cournot. We substitute equations (8) and (9) into equations (5) and (6), and then, the equilibrium solution in the Stackelberg–Cournot game is obtained as the following form: We consider two firms think with different strategies to decide their outputs for profit maximization. Duopoly Model # 1. His duopoly model consists of two firms marketing a homogenous good. 3. Stackelberg model is a leadership model that allows the firm dominant in the market to set its price first and subsequently, the follower firms optimize their production and price. By contrast, our paper focuses on a Stackelberg–Cournot game with imperfect rationality. For simplicity here we consider as duopoly situation, as in Cournot’s model. However, after adding the controlling factor to the chaotic state, the complex situation could be forced to become steady. What is the Stackelberg Model? Seller В also recognises interdependence and realizes that by selling EA output at a higher price OP1 he will share the monopoly profit. It is a duopoly model similar to the duopoly model developed by Joseph Bertrand, in which two firms producing the same good compete in terms of prices. Suppose in Figure 2 seller В thinks that seller A has raised the price of his product, so В will follow him by raising the price of his product. The oldest determinate solution to the duopoly problem is by the French economist, A.A. Cournot in 1838, who took the case of two mineral water springs situated side by side and owned by two firms A and B. Before publishing your Articles on this site, please read the following pages: 1. In this paper, a duopoly Stackelberg model has been proposed. We are committed to sharing findings related to COVID-19 as quickly as possible. Ma and Ren  analyzed a dynamic Cournot–Stackelberg model, which involved a feedback regulation system with one manufacturer and two retailers in the market. Common models that explain oligopoly output and pricing decisions include cartel model, Cournot model, Stackelberg model, Bertrand model and contestable market theory. However, in the Cournot solution the output (OF) is greater than it would be under monopoly (OA). Firms have to select outputs (capacity) in order to maximize profits. The main defect in Cournot’s solution is that each seller assumes his rival’s supply fixed, despite repeatedly observing changes in it. Under the assumption that R&D spillovers only flow from the R&D leader to the R&D follower, a duopoly Stackelberg–Cournot game with heterogeneous expectations is considered in this paper. With the help of it, players can forecast their outputs in a short term. Stackelberg Model: Stackelberg’s equilibrium is mainly based on Stackelberg’s theory of competition, which tells us that two or more companies compete in order to completely dominate the market. (8) Each seller decides about the quantity he wants to produce and sell in each period. There are different diagrams that you can use to explain 0ligopoly markets. Stackelberg duopoly, also called Stackelberg competition, is a model of imperfect competition based on a non-cooperative game. At this price, the total output of mineral water is OF, which is equally divided between the two firms. 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 The Stackelberg Model 3. The Stackerlberg disequilibrium which results from the attempts at leadership on the part of both the sellers is also based on wrong reasoning and arbitrary assumptions. First, the equilibrium quantity is ultimately determined by firms’ R&D spillover, TIE, and marginal cost in a perfectly rational duopoly market consisting of the successive R&D stage and simultaneous production stage, not the R&D input, which is different from our common sense that the Nash equilibrium output is directly related to R&D input [29, 32]. The Chamberlin solution involves a kind of agreement between the two sellers. In reality, the quantity outputs of firms acutely fluctuate when bifurcation and chaos occur; therefore, it is hard for the players to forecast their outputs and make decisions in the future. The Chamberlin Model. The R&D investments before the Cournot production phase have been solved by backward induction. But the price under monopoly (OP) would be higher than under the Cournot solution (OP2). The marginal costs of both A and В are assumed to be zero so that they coincide with the horizontal axis. KJQN202000832), High-Level Talents Program of Chongqing Technology and Business University (Grant No. We assume the follower is a naïve player, he computes his output according to the reaction function, which is derived from equation. Thus the assumption on which this analysis is based is arbitrary and incorrect. But Chamberlin assumes their interdependence. , hold:The abovementioned inequalities of , , and define a region where the Nash equilibrium point is locally stable. Thus there is no equilibrium situation. So seller A does not react to B’s move and compromises with the existence of B. (4) If both A and В desire to be followers: There is a determinate solution because each acts as a follower knowing that the other will also act as a follower. Because the simulation graph in the paper is based on the virtual data under certain conditions, the data used to support the findings of this study are available from the corresponding author upon request. Figure 1 presents a bifurcation diagram of system (13) in the plane when . Firms are identical and produce an homogenous product. Obviously, is a boundary equilibrium point and is the unique Nash equilibrium point. The total output OD1 will be divided between A and В equally as OA and AD1 Notice that in the Cournot solution, the price OP2 exceeds the zero marginal cost and zero prices under perfect competition, and the output OF is less than OD1 under perfect competition. Sweezy’s Kinked Demand Model. The intersection-point equilibria result from a mutual attempt to follow the rival’s leadership. It is a closed model because it does not allow entry of firms. The inverse market demand function is 700 3 P Q = − The marginal cost is constant and equal to $100 for both firms, and there are no fixed costs. Similarly, for any specified value of A’s output, the corresponding value of B’s output maximises B’s profit. Seller В enters the market after him and considers SD1 segment of the market demand curve (DD1) as his demand curve. Before a nonlinear duopoly model is presented, results based on static linear models will be briefly reminded . He will then choose to play whatever role brings him greater profits. But “the theory is developed around the use of reaction functions expressing individual profit maximisation for given values of the rival’s variable.” This limits the use of the theory by excluding the problem of collusion and co-ordination among duopolists. The two firms make simultaneous decisions. The following points highlight the top three models of duopoly. The cournot model has been criticised on the following grounds: 1. The Jacobian matrix at has the following form:The characteristic equation of the matrix iswhere is the trace and is the determinant of the Jacobian matrix ; hence,since ; this means that there are two real roots in the characteristic equation.As we know from the stability theory, the sufficient and necessary conditions for the local stability of Nash equilibrium are that the eigenvalues of Jacobian matrix are inside the unit circle in the complex plane, and it is true only if following Jury’s conditions, Peng et al. This model was developed by the German economist Heinrich von Stackelberg and is an extension of Cournot’s model. Ever since D’Aspremont and Jacquemin proposed AJ model , where completely rational duopoly firms play a two-stage game with Cournot R&D and Cournot production, many papers have studied the influence of technology spillover on enterprise competition and cooperation [28, 39], and imperfect rationality plays an important role in dynamic analysis of R&D spillovers [29–32]. In this paper, the duopoly Stackelberg–Cournot game is divided into two stages. In simple words, let us assume a market with three players – A, B, and C. 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. (10) At the same time, each seller takes the supply or output of its rival as constant. The Cournot Model 2. 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. In order to show bifurcations and chaos, the maximum Lyapunov exponent is also plotted in Figure 1, where positive values show that the chaotic behaviors and the maximum Lyapunov exponent equal to zero at bifurcation point. It describes the strategic behaviour of industries in which there is a dominant firm or a natural leader and the other firms are the followers. 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. Especially, results indicate that small value of R&D spillovers or big value of output adjustment speed may yield bifurcations or even chaos. Then В will follow him and so on till both reach the equilibrium point E. Thus the followership solution is determinate. A. Hołyst and K. Urbanowicz, “Chaos control in economical model by time-delayed feedback method,”, Y. Moreover, the cost reduction represents the R&D production function, characterized by the inverse mapping of the R&D cost function used by D’Aspremont and Jacquemin , with and . With sequential play at the Stackelberg R&D stage, due to R&D’s one-way flow, the cost reduction accruing to the firm 1 just depends on his own investments , and the marginal cost for the R&D leader is given by (the assumption same as that of , while that for the R&D follower is given by . However, there is no doubt that asymmetry information exists widely in production practice. The Chamberlin model is not free from certain weaknesses: 1. Stackelberg’s Duopoly 5. Stage 2 is the Cournot production phase where the strategy space is the choice of output, and in this phase, the choices of R&D investments made in stage 1 are common knowledge, the two oligarchs decide their outputs simultaneously. TOS4. The price is zero because the marginal cost is zero. 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