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It can even be used for aromatherapy and skin care treatments as well. Cournot duopoly, also called Cournot competition, is a model of imperfect competition in which two firms with identical cost functions compete with homogeneous products in a static setting. The strategies are decided in terms of prices rather than quantities. The Nash equilibrium is much different. The Cournot model assumes that firms pick quantities rather than prices. That means an auctioneer chooses the price to equate supply and demand.
The earliest duopoly model was developed in It is developed by the French economist Augstin Cournot. He has noted that this game has a unique equilibrium when demand curves are linear. The Cournot game has a continuous strategy space even without mixing. If a game has a continuous strategy set then it is not always easy to depict the strategic form and outcome matrix is an extensive form as a tree.
In order to present Cournot game, new notation will be useful. The Cournot game model is a duopoly in which two firms chooses output levels in competition with each other. Total output is the monopoly output and it maximizes the sum of payoffs.
It implies a total output of 60 and a price of In order to decide how much of that output of 60 should be produced by each firm. Such output would be a zero sum co-operative game. It is an example of bargaining between firms.
But since the Cournot game is non-co-operative game. It is not necessarily equilibrium despite their Pareto optimality. Each firm produces about quantity he wants to produce and unaware about rivals plan of production. In order to find Nash equilibrium in Cournot game, we need a reaction function. If Brydox produces output then Apex would produce the monopoly output of It is given in following equation with respect to his strategy qa. The reaction function of the two firms is labeled as Ra and Rb in figure.
They cross point c which is the Cournot-Nash-Equilibrium. It is also the Nash equilibrium when the strategies consist of quantities. The equilibrium price is also 40, co-incidentally. In the Cournot game, the Nash equilibrium has the particularly property of stability. If we assume that the initial strategy combination is point x in figure, then it moves the profile closer to equilibrium.
But this is special to the Cournot game and Nash equilibrium is not always stable in this way. In Nash equilibrium, Apex believes that if he changes qa Brydox will not respond by changing qb. Stackelberg equilibrium differs from Cournot equilibrium. For instance, in duopoly market, one firm competes in a dominant position, and it chooses output as decision variable while the other one is in disadvantage, and it chooses price as decision variable in order to gain more market share.
As we have known so far, Bylka and Komar [ 9 ] and Singh and Vives [ 10 ] are the first authors to analyze duopolies, where one firm competes on quantities and the other on prices. Recently, C. Tremblay and V. Tremblay [ 14 ] analyzed the role of product differentiation for the static properties of the Nash equilibrium of a Cournot-Bertrand duopoly. Naimzada and Tramontana [ 8 ] considered a Cournot-Bertrand duopoly model, which is characterized by linear difference equations.
They also analyzed the role of best response dynamics and of the adaptive adjustment mechanism for the stability of the equilibrium. In this paper, we set up a Cournot-Bertrand duopoly model, assuming that two firms choose output and price as decision variable, respectively, and they all have bounded rational expectations. The gaming system can be described by nonlinear difference equations, which modifies and extends the results of Naimzada and Tramontana [ 8 ], which considered the firms with static expectations and described by linear difference equations.
The research will lead to a good guidance for the enterprise decision-makers to do the best decision-making. The paper is organized as follows the Cournot-Bertrand game model with bounded rational expectations is described in Section 2.
In Section 3 , the existence and stability of equilibrium points are studied. Dynamical behaviors under some change of control parameters of the game are investigated by numerical simulations in Section 4. Finally, a conclusion is drawn in Section 5. We consider a market served by two firms and firm produces good ,. There is a certain degree of differentiation between the products and. Firm 1 competes in output as in a Cournot duopoly, while firm 2 fixes its price like in the Bertrand case.
Suppose that firms make their strategic choices simultaneously and each firm knows the production and the price of each other firm.
The inverse demand functions of products of variety 1 and 2 come from the maximization by the representative consumer of the following utility function: subject to the budget constraint and are given by the following equations the detailed proof see [ 15 ] : where the parameter denotes the index of product differentiation or product substitution. The degree of product differentiation will increase as. Products and are homogeneous when , and each firm is a monopolist when , while a negative implies that products are complements.
Assume that the two firms have the same marginal cost , and the cost function has the linear form: We can write the demand system in the two strategic variables, and : The profit functions of firm 1 and 2 are in the form:.
We assume that the two firms do not have a complete knowledge of the market and the other player, and they build decisions on the basis of the expected marginal profit. If the marginal profit is positive negative , they increases decreases their production or price in the next period; that is, they are bounded rational players [ 5 , 15 , 16 ].
The system 6 has four equilibrium points: where ,. Otherwise, there will be one firm out of the market. In order to investigate the local stability of the equilibrium points, let be the Jacobian matrix of system 6 corresponding to the state variables , then where ,.
The stability of equilibrium points will be determined by the nature of the equilibrium eigenvalues of the Jacobian matrix evaluated at the corresponding equilibrium points. Proposition 1. The boundary equilibria , , and of system 6 are unstable equilibrium points when. For equilibrium , the Jacobian matrix of system 6 is equal to These eigenvalues that correspond to equilibrium are as follows: Evidently , then the equilibrium point is unstable.
Also at the Jacobian matrix becomes a triangular matrix These eigenvalues that correspond to equilibrium are as follows: When , evidently. So, the equilibrium point is unstable. Similarly we can prove that is also unstable.
From an economic point of view we are more interested to the study of the local stability properties of the Nash equilibrium point , whose properties have been deeply analyzed in [ 14 ].
The Jacobian matrix evaluated at the Nash equilibrium point is as follows. The trace and determinant of are denoted as and , respectively. Also, we can learn more about the stability region via numerical simulations. In order to study the complex dynamics of system 6 , it is convenient to take the parameters values as follows: Figure 1 shows in the parameters plane the stability and instability regions.
From the figure, we can find that too high speed of adjustment will make the Nash equilibrium point lose stability. We also find that the adjustment speed of price is more sensitive than the speed of output, and when about , the Nash equilibrium point will lose stability, while about the Nash equilibrium point will do that.
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