Chaotic dynamics in Bertrand model with technological innovation

Fenglian Wang1 , Banglun Wang2 , Rongjian Xie3

1, 3College of Management Engineering, Anhui Polytechnic University, Wuhu Anhui, 241000, P. R. China

2School of Mechanical and Automotive Engineering, Anhui Polytechnic University, Wuhu Anhui, 241000, P. R. China

3The School of Management, University of Science and Technology of China, Hefei Anhui, 230000, P. R. China

1Corresponding author

Vibroengineering PROCEDIA, Vol. 15, 2017, p. 134-140. https://doi.org/10.21595/vp.2017.19363
Received 30 October 2017; accepted 6 November 2017; published 1 December 2017

Copyright © 2017 JVE International Ltd. 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.
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Abstract.

In this paper, the dynamics of a Bertrand duopoly game with technology innovation have been studied, which contains boundedly rational and naive players. They have been analyzed that the stability of the equilibrium point, the bifurcation and chaotic behavior of the dynamic system. It has been proved that technology innovation has played a very important role in the stability of Nash equilibrium point. Technology innovation can enlarge the stability region of the speed and control the chaos of the dynamic system effectively.

Keywords: Bertrand duopoly games, technology innovation, nash equilibrium point, chaos.

1. Introduction

Cournot game [1] and Bertrand game [2] are the classic models of oligopoly competition. In Cournot game, oligopoly enterprises put the product output as the decision variable and choose the optimal output for high profits. In Bertrand competition, duopoly enterprises play a game based on product differentiation and determine a proper price to obtain the huge profits. In reality, oligopoly enterprises make the output and price decision dynamically and often adjust the decision based on market demand, the decision of competitors, their own production capacity and so on.

In the past, a large number of literatures did the research on dynamic behavior of Cournot and Bertrand game, such as A.A. Elsadany [3], Xiaolong Zhu [4], H. N. Agiza [5], A. K. Naimzada [6], Jixiang Zhang [7], Luciano Fanti [8], Baogui Xin [9] and so on, which involved several adjustment rules: naïve [8, 10, 11], adaptive [5, 12], bounded rational [3, 4, 7, 13, 14] and local monopolistic approximation [3, 14]. Basically, research results of all these papers show that bifurcation and chaos exist in the dynamic system of Cournot and Bertrand game [3-14]. The parameter, adjustment speed of bounded rational player is an important factor which influences the stability of Nash equilibrium point and incurs bifurcation and chaos. What’s more, keeping low adjustment speed can control the chaos.

Previous research conclusions seem unified but not rich. It still lacks studies about the influence of other parameters on the stability of dynamic system. It is well-known that technology innovation plays an important role in economic development. It can enhance the competitive advantage and increase profit of enterprises. Then, can it improve the stability of dynamic systems?

Following the method of Zhang [7] and Agiza [5], this article explores the dynamics of Bertrand duopoly game between boundedly rational and naive players and study the effects of technology innovation on the dynamics of Bertrand model. The paper is structured as follows: In Section 2, a dynamic Bertrand duopoly game model was built which is composed of players that produce heterogeneous products and have different price adjustment rules. In Section 3, the dynamic behaviors and the equilibrium points were studied. The conditions for the existence and local stability of the equilibrium points will be also analyzed. In Section 4, the dynamic system was simulated via many bifurcation figures. The Section 5 drew the conclusion.

2. The model

We consider a Bertrand-type duopoly market where two oligopolies choose different prices for their heterogeneous products. Players can decide the prices according to the adjustment rules.

Let pi(t), i= 1, 2 represents the price of firm i at discrete-time periods t= 0, 1, 2, ..., Qi represents the output. Following Zhang [7], suppose the market demand function of the players is:

(1)
Q i = a - 2 b p i + d p j ,    

where a>0, b>0,i, j=1, 2, ij . The parameter d measures the degree of substitution of the two products. Large d represents big degree of substitution.

Positive parameter Ai is the initial marginal cost of firm i. The production cost will be reduced by technological innovation. Let xi represents the reduction degree of the marginal cost of the firm i. xi is positive correlation with technology innovation investment and can be used to measure the degree of technology innovation. Furthermore, the firm i can benefit from another firm’s innovation because technique innovation has externality and spillover effect. Let β[0,1] is the degree of technology spillover. The marginal cost function of the players can be assumed as follows:

(2)
c i = A i - x i + β x j ,

where, ci 0 namely xi+βxjAi hold.

With these assumptions, the profit of the firm i in the single period can be given by:

(3)
π i = p i - c i   Q i = p i - A i + x i + β x j a - 2 b p i + d p j .

From the profit maximization by player i, the marginal profits in period t are obtained as:

(4)
π i p i = a - 2 b p i + d p j + b A i - x i + β x j .

Then, the optimal price response function of firm i can be given by:

(5)
p i = a + d p j + b [ A i - ( x i + β x j ) ] 2 b .

Information in the market usually is incomplete. Supposing players use different expectations to adjust the prices. Following Zhang [7]and Agiza [5], suppose player 1 is boundedly rational [7] and player 2 is naïve [5].

Boundedly rational player 1 makes its price decision based on an estimate of the marginal profit π1/p1 [7]. Namely it decides to increase its price p1 if it has a positive marginal profit, or decreases its price when the marginal profit is negative. Then the dynamical equation of player 1 can be given by:

(6)
p 1 t + 1 = p 1 t + k p 1 t π 1 p 1 t ,

where k is a positive parameter which reflects the speed of price adjustment.

Naive player 2 makes its price decision according to the naive expectations rule [8]. The player 2 decides its prices with his reaction function a+dp1+b[A2-(x2+βx1)]/2b. Hence the dynamic equation of the naive expectation player 2 can be given by:

(7)
p 2 t + 1 = a + d p 1 ( t ) + b [ A 2 - ( x 2 + β x 1 ) ] 2 b .

With above assumptions, the duopoly game with heterogeneous players is formed from combining Eqs. (6) and (7). Then the dynamical system of the heterogenous players is described as:

(8)
p 1 t + 1 = p 1 t + k p 1 t [ a - 2 b p 1 ( t ) + d p 2 ( t ) + b ( A 1 - x 1 - β x 2 ) ] , p 2 t + 1 = a + d p 1 ( t ) + b ( A 2 - x 2 - β x 1 ) 2 b .

3. Nash equilibrium and local stability

In this part the equilibria points of dynamic system will be first studied Eq. (8), and then the stability will be discussed.

The dynamic duopoly game will achieve a Nash Equilibrium at last. The possible equilibrium point of the map Eq. (8) can be obtained as nonnegative solution of the nonlinear algebraic system:

(9)
p 1 a - 2 b p 1 + d p 2 + b ( A 1 - x 1 - β x 2 ) = 0 , p 2 = a + d p 1 + b ( A 2 - x 2 - β x 1 ) ] 2 b .

Find that the system (9) is not associated with the parameter k. After the calculation of the system it was found that the map has two equilibrium points:

(10)
E 1 = 0 , p 2 0 ,       E 2 = p 1 * , p 2 * ,

where:

(11)
p 2 0 = a + b A 2 - x 2 - β x 1 2 b ,
p 1 * = 2 b a + b ( A 1 - x 1 - β x 2 ) + d [ a + b ( A 2 - x 2 - β x 1 ) ] 4 b 2 - d 2 ,
p 2 * = 2 b a + b ( A 2 - x 2 - β x 1 ) + d [ a + b ( A 1 - x 1 - β x 2 ) ] 4 b 2 - d 2 .

In the traditional economic view, non-negative equilibrium is meaningful. Obviously, E1 is a boundary equilibria (p20>0). E2 is the unique Nash equilibrium point and has economic meaning provided that:

(12)
2 b a + b ( A 1 - x 1 - β x 2 ) + d [ a + b ( A 2 - x 2 - β x 1 ) ] > 0 , 2 b a + b ( A 2 - x 2 - β x 1 ) + d [ a + b ( A 1 - x 1 - β x 2 ) ] > 0 , 2 b > d ,

where, the above two inequalities are obvious, then Eq. (12) is equivalent to 2b>d.

In order to study the local stability of equilibrium, the Jacobian matrix of map Eq. (8) should be considered. The matrix form is as follows:

(13)
J E = 1 + k [ a - 4 b p 1 + d p 2 + b ( A 1 - x 1 - β x 2 ) ] d k p 1 d 2 b 0 .

The equilibrium point is stable only when all eigenvalues λi (i= 1, 2) of the Jacobian matrix satisfy λi< 0. According to this theory, the following result about E1 can be received.

Proposition 1. The equilibrium point E1 of system Eq. (8) is a saddle point.

Proof. The Jacobian matrix of E1 has the form:

(14)
J ( E 1 ) = 1 + k [ a + d p 2 0 + b ( A 1 - x 1 - β x 2 ) ] 0 d 2 b 0 .

Its’ eigenvalues are:

λ 1 = 1 + k [ a + d p 2 0 + b ( A 1 - x 1 - β x 2 ) ] ,       λ 2 = 0 .

For the condition that a, b, d, k, ci are all positive parameters, λ1> 1 is workable. Then the equilibrium point E1 is a saddle node. The proof of the proposition is completed.

Next the local stability of the Nash equilibrium point E2 will be studied. The Jacobian matrix of E2 is:

(15)
J E 2 = 1 + k [ a - 4 b p 1 * + d p 2 * + b ( A 1 - x 1 - β x 2 ) ] d k p 1 * d 2 b 0 ,

where, the trace of J(E2) is:

(16)
T = T r J E 2 = 1 + k a - 4 b p 1 * + d p 2 * + b ( A 1 - x 1 - β x 2 ) .

The determinant of JE2 is:

(17)
D = D e t J E 2 = - d 2 k p 1 * 2 b .

The characteristic equation of J(E2) is:

(18)
P λ = λ 2 - T λ + D = 0 .

The discriminant is:

(19)
= T 2 - 4 D .

Since =T2+2d2kp1*/b> 0, the eigenvalues of Nash equilibrium E2 are real.

Necessary and sufficient conditions for local stability of the Nash equilibrium E2 are the Jury’s condition, which is given by:

(20)
I : 1 + T + D > 0 , ( II ) : 1 - T + D > 0 , ( III ) : 1 - D > 0 .

Since:

(21)
1 - T + D = - k a - 4 b p 1 * + d p 2 * + b ( A 1 - x 1 - β x 2 ) - d 2 k p 1 * 2 b .

Then replace p1*, p2* Eqs. (21) can be simplified as:

2 k b a + b ( A 1 - x 1 - β x 2 ) + k d [ a + b ( A 2 - x 2 - β x 1 ) ] 2 b > 0 ,

(II) is always satisfied.

Since:

(22)
1 - D = 1 + d 2 k p 1 * 2 b > 0 ,

(III) is always satisfied.

Then focus on the inequality (I).

Since:

(23)
1 + T + D = 2 + k a + b ( A 1 - x 1 - β x 2 - k ( 8 b 2 + d 2 ) p 1 * 2 b + k d p 2 * .

Then replace p1*, p2*. Since, 2b>d (I) is equivalent to:

(24)
4 b 4 b 2 - d 2 - k 4 b 2 + d 2 [ 2 b a + b c 1 + d a + b c 2 ] > 0 .

From what has been mentioned above, the following conclusion can come out:

Proposition 2. The Nash equilibrium at E2 is stable if and only if the inequality Eq. (24) holds.

Proposition 2 characterizes the stability region in which the Nash equilibrium E2 is local stable. The violation of the inequality Eq. (24) will lead to a flip bifurcation [3].

Noticing that the stability region is associated with xi and β. The propositions can be given about the degree of technology innovation xi and the degree of technology spillover β.

Proposition 3. When x1>x10, the evolution of price system Eq. (8) is in a stable state and E2 is the Nash equilibrium point. Otherwise, the price evolution is in bifurcation or chaos. Where:

(25)
x 1 0 = 2 b a + b ( A 1 - β x 2 ) + d [ a + b ( A 2 - x 2 ) ] - 4 b 4 b 2 - d 2 k 4 b 2 + d 2 2 b 2 + d b β .

Proof. According to the stability theory of Jury’s condition, the flip bifurcation occurs when 1+T+D= 0. Namely:

(26)
4 b 4 b 2 - d 2 - k 4 b 2 + d 2 2 b a + b c 1 + d a + b c 2 = 0 .

Then x1=x10.

So, the system is in stable when x1>x10, otherwise in bifurcation or chaos.

Proposition 4. When x2>x20, the evolution of price system Eq. (8) is in a stable state and E2 is the Nash equilibrium point. Otherwise, the price evolution is in bifurcation or chaos. Where:

(27)
x 2 0 > 2 b a + b ( A 1 - x 1 ) + d [ a + b ( A 2 - β x 1 ) ] - 4 b 4 b 2 - d 2 k 4 b 2 + d 2 2 b 2 β + d b .

Proof. According to the stability theory of Jury’s condition, the flip bifurcation occurs when 1+T+D= 0. Namely:

(28)
4 b 4 b 2 - d 2 - k 4 b 2 + d 2 2 b a + b c 1 + d a + b c 2 = 0 .

Then x2=x20.

So, the system is in stable when x2>x20, otherwise in bifurcation or chaos.

From the above description and Proposition, it can be concluded that high technology innovation is beneficial to obtain a steady state and the Nash equilibrium profit. It can expand the stable region and enhance the stability of the product price of market to increase technical innovation.

4. Numerical simulations

The purpose of this part is to illustrate the qualitative behavior of the solutions of the duopoly dynamic system Eq. (8) and provide some numerical evidences to prove above results.

In Fig. 1 (a= 5, b= 1, d= 0.3, A1= 2, A2= 3, x2= 2, β= 0.5, k= 0.32) Nash equilibrium is locally stable approaches to the stable point (p1*,p2*)= (3.031, 3.229) for large values of x1, to be specific, when x1> 0.905. With the reduction of x1, the Nash equilibrium point becoming instable, period-halving bifurcation and chaos will occur.

In Fig. 2 (a= 5, b= 1, d= 0.3, A1= 2, A2= 3, x1= 1, β= 0.5, k= 0.32) the Nash equilibrium is locally stable only when x2> 1.769 (a= 5, b= 1, d= 0.3, A1= 2, A2= 3, x1= 1, β= 0.5, k= 0.32). The dynamic system is in bifurcation or chaos if the technology innovation degree x2 is small.

Fig. 1. Bifurcation diagram with respect to x1

 Bifurcation diagram with respect to x1

Fig. 2. Bifurcation diagram with respect to x2

Bifurcation diagram with respect to x2

5. Conclusions

This paper established the price dynamic game model and then analyzed the influence of technology innovation on the equilibrium stability. The results show that technology innovation plays an important role in improving the stability of equilibrium. Specifically, it can enlarge the stability region and make the original bifurcation and chaos change into stability to increase the degree of technology innovation.

Acknowledgements

We wish to thank Hua Zhao and Jianjun Long and some commissions for the research assistance. This study is supported by National Social Science Fund of China (16BGL201), National Social Science Fund of China (14BGL163), Philosophy Social Science Planning Project of Anhui Province (AHSKY2016D14), Research Start-up Fund of Anhui Polytechnic University (2015YQQ004), Youth Research Fund of Anhui Polytechnic University (2016YQ32).

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