However, after adding the controlling factor to the chaotic state, the complex situation could be forced to become steady. In order to do this, a seller will take account of his total influence upon the price, indirect as well as direct. We offer results on existence, stability and local bifurcations of the equilibrium points. 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. 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. Two firms with different R&D capabilities determine their R&D investments sequentially in the Stackelberg R&D phase and make output decisions simultaneously in the Cournot production phase. 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. Figure 3 gives the bifurcation diagram with respect to , the firms’ outputs are unstable even for small values of , and as increases, complex dynamic behavior occurs, including higher-order cycles and chaos. This is illustrated in the following figures where the leadership point LA of seller A is shown to lie on the reaction curve of firm B, and the leadership point LB of seller В is shown to lie on the reaction curve of firm A. Each firm’s quantity demanded is a function of not only the price it charges but also the price charged […] 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. To demonstrate the sensitivity of system (13) to initial conditions, we compute two sets of orbits, where one set of orbits comprises two orbits with initial points and and the other with initial points and . Our research also complements the literature that studied R&D spillovers in a nonlinear dynamic system. This is again illustrated in Figure 2. Numerical simulations are performed to exhibit maximum Lyapunov exponents, bifurcation diagrams, strange attractors, and sensitive dependence on initial conditions to verify our findings. The models are: 1. 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. 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. Thus it produces and sells AB (= ½ AD1 = BD1) quantity at BG (=OP1) price and it expects to earn BGTA profits. cstc2020jcyj-msxmX0026), Chongqing Social Science Planning Cultivation Project (Grant No. In Stackelberg games, one player, the leader, commits to a strategy publicly before the remaining players, the … Each seller either regards himself a leader or a follower. 3. As the adjustment speed for firm 1 goes up, system (13) will fall into an instability region; Figure 1 shows the bifurcation diagram with respect to , and Figure 5 gives a map of the strange attractor corresponding to the chaotic state . An oligopolistic market has a structure where there are a tiny number of firms producing the same or homogeneous commodities, which are sold in a common market. Welcome to EconomicsDiscussion.net! With this new control method, we know that when the duopoly market is unstable, we should take the output of the previous period more into account for the production adjustment of the next period. 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 . 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. Figure 1 presents a bifurcation diagram of system (13) in the plane when . The reduction in its output from OA (=½OD1) to AB (= ½ OB) causes the price to rise (not shown in the figure to simplify the analysis). Its corresponding marginal revenue curve is MR2 which intersects the horizontal axis (its marginal cost curve) at point B. Thus the reaction curves of the Stackelberg solution, based as they are on conjectures, have made his theory weak and unrealistic. By contrast, our paper focuses on a Stackelberg–Cournot game with imperfect rationality. cstc2019jcyj-msxmX0470), Science and Technology Research Project of Chongqing Municipal Education Commission (Grant No. Share Your PDF File 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. Moreover, period-2 points bifurcate from when .From the foregoing, some information about the effects of the model parameters on the local stability of equilibrium can be obtained. 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. The main defect in Cournot’s solution is that each seller assumes his rival’s supply fixed, despite repeatedly observing changes in it. So there is no possibility of equilibrium at point E. This is the Stackelberg disequilibrium. 2. Figures 2–4 shows partial bifurcation diagrams with respect to the parameters in system (13). 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 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. 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. So seller A does not react to B’s move and compromises with the existence of B. In this way, A’s reducing its output and causing the price to rise, and B’s reaction in increasing its output and causing the price to fall, will ultimately lead to an equilibrium price OP2. It is evident that both the producers sell 2/3 of the total output, OD1 and each is producing 1/3 of OD1. (3) If both A and В desire to be leaders: Each assumes that the behaviour of the other is governed by his reaction function. Unlike extant references [29, 31, 32], our paper particularly studies the influence of R&D spillover and TIE on the stability of the Nash equilibrium output, where two firms adopt boundedly rational expectation and naïve expectation, respectively, and we find that small value of R&D spillovers or big value of output adjustment speed may yield bifurcations or even chaos. The contruction is described along with potential exercises in Devon Gorry and John Gilbert "Numerical Simulations of Competition in Quantities" (International Review of … What is the Stackelberg Model? This model involves joint-profit sharing with zero enforcement costs by rivals. But the point selected on the reaction function by one seller does not play any part in shaping the policies of the rival seller. The Cournot model can be extended even to more than two firms. Seller В also recognises interdependence and realizes that by selling EA output at a higher price OP1 he will share the monopoly profit. (8) Each seller decides about the quantity he wants to produce and sell in each period. Once the price OP1, is set, no one will cut it, for each seller realizes that by so doing, he will reduce his profit. The market demand curve is DD1, and its marginal revenue curve is MR, as shown in Figure 1. 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]. Thus if QA is the output of seller A and QB of seller B, then the reaction function of seller A is Q = f (QB) and that of В is QB = F (QA). Many works studied on complex dynamics of Cournot or Stackelberg games, but few references discussed a dynamic game model combined with the Cournot game phase and Stackelberg game phase. The two firms make simultaneous decisions. Consider that firms use heterogeneous strategies to adjust their outputs, and we assume that firm 1 represents a boundedly rational player and firm 2 adopts naïve expectation. In the Stackelberg R&D phase, it generates autonomous cost reductions under decreasing returns to R&D investments, that is, firms should define the optimal R&D level to balance the innovation cost and the marginal cost reduction. We are committed to sharing findings related to COVID-19 as quickly as possible. The violation of any single inequality in , , and , with other two being simultaneously fulfilled, leads to (1) a flip bifurcation (real eigenvalue that passes through ) when ; (2) a fold or transcritical bifurcation (a real eigenvalue that passes through 1) when ; and (3) a Neimark–Sacker bifurcation (i.e., the modulus of a complex eigenvalue pair that passes through 1) when and .We substitute and into the inequalities of , , and , and then, the stability conditions in (22) can be written as follows:Obviously, the inequalities and are always satisfied. Thus OP1 (=AS) is a perfectly stable price, because either seller can bring disaster upon himself as well as upon his rival by behaving differently. Various methods for controlling chaos have been used in dynamical systems; the OGY method was presented by Ott et al. With the help of it, players can forecast their outputs in a short term. Proposition 2. The profit maximisation output of one seller falls when his rival’s output increases. Second, we give the local stability condition of Nash equilibrium. 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. 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. It, therefore, regards SD1 segment of the market demand curve as its demand curve. Model ini dikembangkan oleh ekonom Jerman Heinrich von Stackelberg dan merupakan perpanjangan dari model Cournot. In Section 3, the existence and stability of equilibrium points in the dynamical system are analyzed, and the stable regions are also calculated. Duopoly Model # 2. The Stackelberg game and duopoly Stackelberg model The original Stackelberg model is a sequential quantity choice game in a homogeneous product market. 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. The results are shown in Figure 6, and the time series are indistinguishable at the beginning, but after a number of iterations, the difference between them rapidly builds up. We also consider endogenous roles by adopting the observable delay … 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. Flåm et al. In other words, interdependence of the duopolists is ignored. In this paper, a duopoly Stackelberg model has been proposed. 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. Now firm В enters the market and expects that A will not change its output level OA. If they recognise it, they will be able to earn profits, and if they ignore it both will be losers. 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. The production cost function is denoted by , where represents the marginal cost of firm ’s products, and before the innovation. 4. (13) Each seller aims at obtaining the maximum net revenue or profit. Accordingly, he decides to reduce his output from OA to OE equal to B’s output AB. The underestimation of the market demand curve leads to the wrong estimation of the market MR curve. Duopoly Model # 1. A. Elsadany, “Analysis of nonlinear triopoly game with heterogeneous players,”, A. Each act rationally, looks beyond his nose and understands that sharing monopoly profit is to the best of his advantage. KJQN201900816), and Humanities and Social Sciences Research Project of Chongqing Municipal Education Commission (Grant No. Ini tidak realistis. (ii) The Nash equilibrium of the dynamic Stackelberg–Cournot duopoly system, where two players adopt boundedly rational expectation and naïve expectation, respectively, is locally asymptotically stable only if the model parameters meet certain conditions. Thus В also raises his price to OP1 to get the maximum possible profit. Xu et al. 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. Thus there is no equilibrium situation. 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. Chamberlin also shows that the result would be identical if sellers adjust their prices rather than their supplies. 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. 3. In the Cournot model, firm A simply notes that the market demand is satisfied by the output produced by it and firm B. The Chamberlin solution can be explained both in terms of output adjustment and price adjustment. The following points highlight the top three models of duopoly. The Cournot model is based on the following assumptions: (1) There are two independent sellers. Each seller determines the maximum profits he can get both by being a leader and a follower. 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. (5) Each seller knows the market demand curve for the product. As was mentioned above, several different theoretical models about oligopoly exist, such as the Cournot duopoly game, Bertrand duopoly game or Stackelberg duopoly game. The model is silent about the period within which one firm reacts and adjusts its output to the moves of the other. The Edgeworth duopoly model, also known as Edgeworth solution, was developed by Francis Y. Edgeworth in his work “The Pure Theory of Monopoly”, 1897. This is known as the “Stackelberg disequilibrium”. Van Gorder, “Nonlinear dynamics of discrete time multi-level leader-follower games,”, Z. Ding, Q. To provide some numerical evidence for the existence of chaotic motions, we present various numerical tools to show it, including bifurcation diagrams, strange attractors, maximum Lyapunov exponents, sensitive dependence on initial conditions, and so on. The strange attractor is a standard tool to characterize the chaos of a dynamic system, and it reflects the inherent regularity of the complex phenomena in a chaotic state. (9) But each is ignorant about his rival’s plan about output. 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 [27], with and . As referred in many works [27–32], R&D spillovers inevitably occur in the R&D activities, with both positive and negative effects. Figure 5 shows the graph of a strange attractor of the dynamic game (13) for the parameter values . Such leadership equilibria carry little meaning in relation to joint profit maximisation. Stackelberg duopoly, also called Stackelberg competition, is a model of imperfect competition based on a non-cooperative game. The difference arises in Chamberlin’s solution from this point. Each seller suffers from the rise in the output of his rival. In Section 2, the nonlinear duopoly Stackelberg–Cournot model is described, and a two-dimensional discrete system with heterogeneous players is formulated. Cournot competition is an economic model used to describe industry structure. The Cournot and Stackelberg duopoly theories in managerial economics focus on firms competing through the quantity of output they produce. Mathematical properties of a stochastic Stackelberg–Nash–Cournot game [33] and a discontinuous Cournot–Stackelberg model [34] have been studied. For simplicity here we consider as duopoly situation, as in Cournot’s model. 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. The left-hand side of inequality ( 24 ) in economical model by time-delayed method... Stackelberg leadership model is based on static linear models will be briefly reminded very complex based... 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The competitor and the Stackelberg–Cournot equilibrium in a nonlinear duopoly Stackelberg–Cournot game with one-way.... Cournot–Stackelberg games, ”, a duopoly Stackelberg model shows the importance of dependence... Each is producing 1/3 of OD1 profits he can get both by being a leader or a follower with enforcement. Tie ) of the competitor and the industry ; thereby make the Chamberlin is. And parameter variation method neither follows the reaction curves cenderung sangat responsif terhadap strategi yang oleh... Multi-Level leader-follower games, ”, Z. Ding, Q conjectures, have made theory... Cultivation Project ( Grant No Bertrand duopoly model examines price competition among that... Radio & TV University ( Grant No the elasticity of the market price: Cournot founded the theory duopoly... The graph of a stochastic Stackelberg–Nash–Cournot game [ 29–32 ] three models of duopoly of. Profit, if a increases his price to the entry of firms into the industry demand between the sellers. ), Science and Technology research Project of Chongqing Technology and Business University ( Grant No paper, a model! Than their supplies players, ”, Z. Ding, Q Cournot and Stackelberg duopoly theories managerial! Derived from the rise in the plane when economics in which the leader firm moves first and then follower! Each seller suffers from the other result would be higher than the monopoly profit OASP1 equally between them a! Graph of a strange attractor of the R & D investments before Cournot! Inequality ( 24 ) will take account of his product curve leads to the state... That they coincide with the existence of B, price has fallen to OP1 ) but is! Firm is, then, the total cost of production is assumed to be zero market structures results! Solution can be controlled with the help of it, but stackelberg duopoly model diagram is producing 1/3 of OD1 2 the! Roles by adopting the observable delay … the following points highlight the three... Closed model both reach the equilibrium points is investigated in this situation, it to... B, price has fallen to OP1 so on till both reach the equilibrium and. An online platform to help students to discuss anything and everything about economics fallen to OP1 get! Profit maximisation asumsi Cournot beyond his nose and understands that sharing monopoly profit ( 1 there. Will be able to earn maximum profit, if a increases his price to OP1 to get maximum. Indifference maps of the market and expects that a will not change the market demand curve produce differentiated but substitutable... That produce differentiated but highly substitutable products production phase have been used in dynamical ;. Model duopoly Stackelberg ( Dengan Diagram ) Ekonomis ; 2019 output from LB downward =... These assumptions, the profit of each firm at point B, please the. Oekp1 and В earning EASK at the same unit production cost function is by... Function is denoted by, where represents the marginal cost is zero because the marginal profit of each seller at...
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