Firms have to select outputs (capacity) in order to maximize profits. 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. It may result from the assumption that the rival seller moves along a reaction curve which, in reality, does not exist for him. They are based on the assumption that the value of the rival’s variable is given regardless of the seller’s own moves. Dynamics of the system under different regimes of the main parameters, such as the R&D leader’s adjustment speed, R&D investments, technology spillovers, and TIE, have been explored. Cournot uses the example of mineral spring water, […] Thus the reaction curves of the Stackelberg solution, based as they are on conjectures, have made his theory weak and unrealistic. The underestimation of the market demand curve leads to the wrong estimation of the market MR curve. The equilibrium is again stable and determinate. Assume that the inverse demand function has the linear form , where the total supply , and the positive constant represents the maximum amount of outputs that can be brought to the market. As discussed in Section 3, many parameters, such as the R&D leader’s adjustment speed, R&D spillover, and TIE, affect the stability of equilibrium output. In the kinked demand curve model, the firm maximises profits at Q1, P1 where MR=MC. Marshall, therefore, regarded Cournot’s model as “incapable of a universal solution.” This is because it is not possible to find an actual duopoly market where each duopolist acts autonomously and output is the ‘sole parameter of action’. The following points highlight the top three models of duopoly. Figures 2–4 shows partial bifurcation diagrams with respect to the parameters in system (13). The price is zero because the marginal cost is zero. If В acts as the leader and increases his output from LA onward, the reaction on A would be to reduce his output from LB downward. Xu et al. Se supone, por von Stackelberg, que un duopolista es lo suficientemente sofisticado como para reconocer que su competidor actúa bajo el supuesto de Cournot. In Stackelberg games, one player, the leader, commits to a strategy publicly before the remaining players, the … (1) Seller A desires to be a leader, and В a follower: In this situation, the duopolists follow consistent behaviour patterns and the solution is determinate. By contrast, our paper focuses on a Stackelberg–Cournot game with imperfect rationality. Suppose firm A is the only producer in which case it produces and sells OA (=½ OD1) quantity when it’s MR, equals its marginal cost curve (horizontal axis) at point A. These papers, which studied on Stackelberg–Cournot games or Cournot–Stackelberg games, are primarily based on perfect rationality. A. Elsadany, H. N. Agiza, and E. M. Elabbasy, “Complex dynamics and chaos control of heterogeneous quadropoly game,”, Z. Ding, Q. Li, S. Jiang, and X. Wang, “Dynamics in a cournot investment game with heterogeneous players,”, A. A. Hołyst and K. Urbanowicz, “Chaos control in economical model by time-delayed feedback method,”, Y. 2. Thus it produces and sells AB (= ½ AD1 = BD1) quantity at BG (=OP1) price and it expects to earn BGTA profits. The two firms make simultaneous decisions. Dalam model duopoli klasik Cournot, kedua pemain menetapkan kuantitas mereka secara independen. 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. Thus in Chamberlin’s model the sellers are independent, yet they are in a kind of collusion which leads to stable equilibrium, a sort of monopoly equilibrium. System (13) undergoes a flip bifurcation at when . With the help of it, players can forecast their outputs in a short term. Model ini dikembangkan oleh ekonom Jerman Heinrich von Stackelberg dan merupakan perpanjangan dari model Cournot. The classic oligopoly model, proposed by Augustin Cournot originally in 1838 [1], gives a mathematical description of the competitions in oligopolistic markets, and it shows how the firms influence each other in making production decisions. However, there is no doubt that asymmetry information exists widely in production practice. It is also shown that the chaotic behaviors can be controlled with the state variables feedback and parameter variation method. Accordingly, assuming that В will continue to sell the same quantity AB (=BD1), it regards the remaining portion of the market OB available to it. In a nonlinear and dynamic discrete production system, many factors, such as the adjustment speed, R&D investments, TIE, and R&D spillovers, will make the market deviate from the equilibrium state and even become chaotic. Jika hanya ada dua pemain, maka masing-masing akan cenderung sangat responsif terhadap strategi yang diambil oleh pesaing. The Nash equilibrium point is stable for any given , which means, if the speed of adjustment of firm 1 is in the interval region defined by , the yields of the two firms will tend towards the Nash equilibrium . A. Elsadany, “A dynamic cournot duopoly model with different strategies,”, S. S. Askar and K. Alnowibet, “Nonlinear oligopolistic game with isoelastic demand function: rationality and local monopolistic approximation,”, S. S. Askar and K. Alnowibet, “Cooperation versus noncooperation: cournot duopolistic game based on delay and time-dependent parameters,”, T. Puu, “On the stability of cournot equilibrium when the number of competitors increases,”, B. C. Snyder, R. A. V. Gorder, and K. Vajravelu, “continuous-time dynamic games for the cournot adjustment process for competing oligopolists,”, B. Xin and T. Chen, “On a master-slave bertrand game model,”, R. Wu and R. A. Duopoly Model # 1. Suppose the price is anywhere between OP1 and OP2 In order to earn maximum profit, if A increases his price to OP1. 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. Stackelberg’s Duopoly 5. Seller В also recognises interdependence and realizes that by selling EA output at a higher price OP1 he will share the monopoly profit. Each duopolist sells 1/3 of the market i.e. This may lead the collusive firms to charge a price higher than the monopoly price. Sweezy’s Kinked Demand Model. This model involves joint-profit sharing with zero enforcement costs by rivals. In this situation, it tries to adjust its price and output. The cournot model has been criticised on the following grounds: 1. The Stackelberg leadership model is a strategic game in economics in which the leader firm moves first and then the follower firms move sequentially. 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. The principal difierence between the Cournot model and the Stack-elberg model is that instead of moving simultaneously (as in the Cournot model) the flrms now move sequentially. Consider an industry with two firms. The literature on Cournot or Stackelberg games has been studied a lot, respectively, but few references discussed a dynamic game model combined with the Cournot game phase and Stackelberg game phase. Stackelberg’s Duopoly Model (With Diagram) Article Shared by Trisha. 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. Various methods for controlling chaos have been used in dynamical systems; the OGY method was presented by Ott et al. From Figure 1, we can see that the orbit of the quantity outputs approaches to the stable fixed point for the adjustment speed ; furthermore, we can get the optimal investments of each firm with and . Kinked Demand Curve Diagram. Each seller either regards himself a leader or a follower. The difference arises in Chamberlin’s solution from this point. Chamberlin also shows that the result would be identical if sellers adjust their prices rather than their supplies. 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. Jianjun Long, Hui Huang, "A Dynamic Stackelberg–Cournot Duopoly Model with Heterogeneous Strategies through One-Way Spillovers", Discrete Dynamics in Nature and Society, vol. Fellner does not agree with Chamberlin that monopoly solution is possible under duopoly interdependence. 20SKJD072). Referring to previous literatures, naïve [5–9], boundedly rational [3, 6–8, 10–15], adaptive [6, 7, 9, 10, 13, 16], local monopolistic approximation [15, 17–19], expectations are commonly available strategies and have been adopted to describe dynamics in Cournot games [6–8, 18, 20–22] and Stackelberg games [3, 15, 23, 24], with two or more players. The Chamberlin Model. (10) At the same time, each seller takes the supply or output of its rival as constant. Firm A finds that with the entry of B, price has fallen to OP1 from OP. A Duopoly Example. We assume the follower is a naïve player, he computes his output according to the reaction function, which is derived from equation. Proposition 1. (3) If both A and В desire to be leaders: Each assumes that the behaviour of the other is governed by his reaction function. This is illustrated in Figure 2 where the “variable” is interpreted as price. 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. The main defect in Cournot’s solution is that each seller assumes his rival’s supply fixed, despite repeatedly observing changes in it. In simple words, let us assume a market with three players – A, B, and C. 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. If they recognise it, they will be able to earn profits, and if they ignore it both will be losers. Recently, a new control method called as control strategy of the state variables feedback and parameter variation was proposed [45] and had been used in the work of [8, 13, 26]. 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. The assumption that each duopolist can act without any output reaction from the other is unrealistic. In our model, the oligopolistic market contains two firms where firm 1 is called as the R&D leader and firm 2 is called as the R&D follower. 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,”. Before a nonlinear duopoly model is presented, results based on static linear models will be briefly reminded . Content Guidelines 2. 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. 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. If A acting as the leader raises his price from LB upward, В follows him by raising his price from LA upward, so that both ultimately reach the equilibrium point E. Thus the solution is determinate. Accordingly, he decides to reduce his output from OA to OE equal to B’s output AB. Its corresponding marginal revenue curve is MR2 which intersects the horizontal axis (its marginal cost curve) at point B. In addition, one-way R&D spillovers [27, 28] can happen on account of the gap of R&D capabilities between firms, that is, R&D spillovers only flow from enterprises with stronger R&D capabilities to weaker ones in the R&D process, and this phenomenon would be applied in our paper. 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. Our mission is to provide an online platform to help students to discuss anything and everything about Economics. Thus it is a static model. In this paper, a duopoly Stackelberg model of competition on output with stochastic perturbations is proposed. (2) They produce and sell a homogeneous product, mineral water. The reason there are more than one model of oligopoly is that the interaction between firms is very complex. However, in the Cournot solution the output (OF) is greater than it would be under monopoly (OA). But the point selected on the reaction function by one seller does not play any part in shaping the policies of the rival seller. Welcome to EconomicsDiscussion.net! This would tend to drive down the price to the competitive level in the long-run. With these assumptions, the profit of each firm is, Then, the marginal profit of each firm at point is given by. Flåm et al. The Cournot Model 2. cstc2020jcyj-msxmX0026), Chongqing Social Science Planning Cultivation Project (Grant No. The Nash equilibrium point is asymptotically stable if . The dynamic of duopoly Stackelberg model with bounded rationality and local approximation has been analyzed. Basic properties of the game have been analyzed by means of bifurcation diagram and strange attractor. Firms are identical and produce an homogenous product. 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. (11) Neither of them fixes the price for its product, but each accepts the market demand price at which the product can be sold. In the Cournot model, firm A simply notes that the market demand is satisfied by the output produced by it and firm B. Each reaction curve expresses the output of each seller as a function of his rival’s output. The market demand curve is DD1, and its marginal revenue curve is MR, as shown in Figure 1. Copyright © 2020 Jianjun Long and Hui Huang. A sells ОС and В sells CF. The duopoly firms A and В in equilibrium charge OP2 price and sell OF output. Chang, and M.-j. Thus A’s reaction function gives the value of QA which maximises A’s profit for any specified value of B’s output (QB). The R&D investments before the Cournot production phase have been solved by backward induction. The Stackelberg game and duopoly Stackelberg model The original Stackelberg model is a sequential quantity choice game in a homogeneous product market. There are based on each duopolist’s desire to maximise his profit from playing both the roles of leadership and followership. Stackelberg Model Graphically(cont): q2 q’ q’’qM q1 Isoprofit = πM =1 single point π’< πM=(1/b)((a-c)/2)^2 Given q2, firm 1 chooses its best response i.e. 6. (7) Both have identical costs and identical demands. Then, condition (23) becomesWe can obtain the threshold given by the vanishing of the left-hand side of inequality (24). A. Elsadany and A. M. Awad, “Dynamical analysis and chaos control in a heterogeneous kopel duopoly game,”, H. N. Agiza and A. 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. This website includes study notes, research papers, essays, articles and other allied information submitted by visitors like YOU. 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]. 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. It thus sells ½OB. In this model, two competing firms, where one is called as the leader and the other is called as the follower, determine their outputs successively [3] and the leader know the follower’s reaction function, so the leader usually gains more profits than the follower for his first-mover advantage. The Cournot solution is unrealistic because it assumes zero cost of production. Now В acts as the leader and A as the follower. 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). TOS4. Each seller determines the maximum profits he can get both by being a leader and a follower. 2020, Article ID 3251609, 11 pages, 2020. https://doi.org/10.1155/2020/3251609, 1School of Management Science and Engineering, Chongqing Technology and Business University, Chongqing 400067, China, 2School of Business, Hunan Agricultural University, Changsha 410128, Hunan, China. This is because the R&D input is also determined by firms’ R&D spillover, TIE, and marginal cost with backward induction. (6) The cost of production is assumed to be zero. At this price, the total output of mineral water is OF, which is equally divided between the two firms. Complexity behaviors, such as period doubling and chaotic attractors, will also occur, when the maximum Lyapunov exponents of the system (13) are positive. It is a closed model because it does not allow entry of firms. Figure 5 shows the graph of a strange attractor of the dynamic game (13) for the parameter values . Thus a change in MC, may not change the market price. The Stackelberg solution successfully links the duopoly problem to a family of related market structures. Unlike [15], we increase the study of the R&D spillover coefficient. (ii) When the duopoly is a bounded rational and adopts a different output adjustment mechanism, is there stable output? Interpreted as the leader firm moves first and then the follower firms move sequentially 1 ) there different. Described, and a discontinuous Cournot–Stackelberg model [ 34 ] have been by! Between OP1 and OP2 in order to earn profits, and Humanities and Social Sciences research of! 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