6. 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. Disclaimer Copyright, Share Your Knowledge Unlike [15], we increase the study of the R&D spillover coefficient. 2019PY41), Science and Technology Research Project of Chongqing Municipal Education Commission (Grant No. 1955046), Science and Technology Research Project of Chongqing Municipal Education Commission (Grant No. The duopoly firms A and В in equilibrium charge OP2 price and sell OF output. We find that: (i) the boundary equilibrium of the dynamic Stackelberg–Cournot duopoly system, where two players adopt boundedly rational expectation and naïve expectation, respectively, is unstable if the Nash equilibrium of the system is strictly positive. The Nash equilibrium point is asymptotically stable if . Its corresponding marginal revenue curve is MR2 which intersects the horizontal axis (its marginal cost curve) at point B. 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. Stackelberg Model. Figures 2–4 shows partial bifurcation diagrams with respect to the parameters in system (13). [15], hold:The abovementioned inequalities of , , and define a region where the Nash equilibrium point is locally stable. Downloadable! 3. Review articles are excluded from this waiver policy. 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. We analyze two different scenarios: (7) Both have identical costs and identical demands. 1. If they enter into a collusive agreement, they may maximise their profits jointly. There are different diagrams that you can use to explain 0ligopoly markets. But the price under monopoly (OP) would be higher than under the Cournot solution (OP2). 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. (10) At the same time, each seller takes the supply or output of its rival as constant. Share Your PDF File It is evident that both the producers sell 2/3 of the total output, OD1 and each is producing 1/3 of OD1. 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. The Cournot and Stackelberg duopoly theories in managerial economics focus on firms competing through the quantity of output they produce. This is again illustrated in Figure 2. 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. [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. The production cost function is denoted by , where represents the marginal cost of firm ’s products, and before the innovation. 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. ADVERTISEMENTS: List of oligopoly models: 1. It thus sells ½OB. Numerical simulations are performed to exhibit maximum Lyapunov exponents, bifurcation diagrams, strange attractors, and sensitive dependence on initial conditions to verify our findings. (3) The total output must be sold out, being perishable and non-storable. Our research also complements the literature that studied R&D spillovers in a nonlinear dynamic system. As a result, its expected profits decline to OP JA. 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 . Research results show that complex dynamic behaviors would occur as model parameters vary, such as cycles and chaos, and we can stabilize the chaotic behavior of the system to a stable fixed point by introducing an appropriate controlling parameter. But A does not follows В because he also considers himself a leader and so does not reduce his output from LB downward. Jianjun Long, Hui Huang, "A Dynamic Stackelberg–Cournot Duopoly Model with Heterogeneous Strategies through One-Way Spillovers", Discrete Dynamics in Nature and Society, vol. Basic properties of the game have been analyzed by means of bifurcation diagram and strange attractor. 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. Assume that rst Firm 1 moves and chooses q 1:In the second stage, after observing q 1;Firm 2 moves and chooses q 2: 4. 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. Each reaction curve expresses the output of each seller as a function of his rival’s output. What is the Stackelberg Model? Duopoly Model # 2. Firms have to select outputs (capacity) in order to maximize profits. 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. It is important to bear in mind, there are different possible ways that firms in Oligopoly can behave. (8) Each seller decides about the quantity he wants to produce and sell in each period. 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). 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. When a seller remains passive to changes in price or output of his rival, it is a direct influence. cstc2019jcyj-msxmX0470), Science and Technology Research Project of Chongqing Municipal Education Commission (Grant No. Prof. Chamberlin proposed a stable duopoly solution recognising mutual dependence between the two sellers. 5. Thus the reaction curves of the Stackelberg solution, based as they are on conjectures, have made his theory weak and unrealistic. Obviously, is a boundary equilibrium point and is the unique Nash equilibrium point. This website includes study notes, research papers, essays, articles and other allied information submitted by visitors like YOU. The market demand curve is DD1, and its marginal revenue curve is MR, as shown in Figure 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. But, as a matter of fact, neither follows the reaction function of the other. Flåm et al. Thus it is a static model. KJQN201804005), Scientific Research Project of Chongqing Radio & TV University (Grant No. 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 . 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. 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. (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. 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. Each firm’s quantity demanded is a function of not only the price it charges but also the price charged […] The Cournot Model 2. This is known as the “Stackelberg disequilibrium”. However, after adding the controlling factor to the chaotic state, the complex situation could be forced to become steady. It was formulated by Heinrich Von Stackelberg in 1934. The two firms make simultaneous decisions. The equilibrium is again stable and determinate. The exclusion of the problem of collusion has led to unrealistic results. It, therefore, regards SD1 segment of the market demand curve as its demand curve. Cournot uses the example of mineral spring water, […] The underestimation of the market demand curve leads to the wrong estimation of the market MR curve. Before publishing your Articles on this site, please read the following pages: 1. 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. They do not sign it, but each seller is intelligent enough to realise the importance of mutual dependence. 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). 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. Privacy Policy3. Each seller either regards himself a leader or a follower. How to draw a duopoly equilibrium summary diagram - YouTube In this sheet we set up a basic Stackelberg duopoly with linear demands and constant marginal costs. Se supone, por von Stackelberg, que un duopolista es lo suficientemente sofisticado como para reconocer que su competidor actúa bajo el supuesto de Cournot. Before a nonlinear duopoly model is presented, results based on static linear models will be briefly reminded . Our paper differs from these aforementioned references in three ways. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Once the price OP1, is set, no one will cut it, for each seller realizes that by so doing, he will reduce his profit. Van Gorder, “Nonlinear dynamics of discrete time multi-level leader-follower games,”, Z. Ding, Q. 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. With the help of it, players can forecast their outputs in a short term. The model is silent about the period within which one firm reacts and adjusts its output to the moves of the other. In the kinked demand curve model, the firm maximises profits at Q1, P1 where MR=MC. TOS4. 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. Thus the assumption on which this analysis is based is arbitrary and incorrect. In this paper, a duopoly Stackelberg model has been proposed. We also consider endogenous roles by adopting the observable delay … Seller В enters the market after him and considers SD1 segment of the market demand curve (DD1) as his demand curve. 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. Now it is A’s turn to follow В in raising the price of his product. 20SKJD072). 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. A high price yielding very high profits may lead to the entry of firms into the industry; thereby make the Chamberlin solution impossibility. But problems might arise in sharing profit by the two sellers. However, in the Cournot solution the output (OF) is greater than it would be under monopoly (OA). The Stackelberg model is a quantity leadership model. 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]. Each act rationally, looks beyond his nose and understands that sharing monopoly profit is to the best of his advantage. Each firm knows its own total cost of production, the total cost of production of the competitor and the industry demand. 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. The Cournot model can be extended even to more than two firms. (2) They produce and sell a homogeneous product, mineral water. (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. 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 firm often underestimates the elasticity of the market demand curve and overestimates the elasticity of its own demand curve. We assume the follower is a naïve player, he computes his output according to the reaction function, which is derived from equation. According to Chamberlin, when interdependence is recognised between sellers, both direct and indirect influences of a change in the price or output of a seller lead to a stable industry equilibrium with monopoly price and output.

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