CrossMark

Available online at www.sciencedirect.com

ScienceDirect

Procedia Economics and Finance 19 (2015) 146 - 153

The Economies of Balkan and Eastern Europe Countries in the changed world, EBEEC 2014, Nis,

Serbia

On the dynamics of a duopoly game with differentiated goods

Georges Sarafopoulos *

Department of Economics, Democritus University of Thrace, Komotini, 69100 Greece.

Abstract

The present study extends Fanti, Gori, 2012 (L. Fanti, L. Gori, 2012. The dynamics of a differentiated duopoly with quantity competition, Economic Modelling, 29, 421-427). In this study we investigate the dynamics of a nonlinear discrete-time duopoly game with differentiated goods, linear demand and quadratic cost functions. The game is modeled with a system of two difference equations. Existence and stability of equilibria of this system are studied. We show that the model gives more complex chaotic and unpredictable trajectories as a consequence of change in the parameter of horizontal product differentiation and a higher (lower) degree of product differentiation (weaker or fiercer competition) destabilize (stabilize) the economy. The chaotic features are justified numerically via computing Lyapunov numbers and sensitive dependence on initial conditions. Also, we show that in the case of quadratic costs there are stable trajectories and a higher (lower) degree of product differentiation does not tend to destabilize the economy.

© 2015TheAuthors.PublishedbyElsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review will be underresponsibility ofDepartment of Accountancy and Finance, Eastern Macedonia and Thrace Institute of Technology, Kavala, Greece.

Keywords: Product differentiation; Discrete Dynamical System; Equilibrium; Stability; Chaotic Behavior. 1. Introduction

An Oligopoly is a market structure between monopoly and perfect competition, where there are only a few number of firms in the market. The dynamic of an oligopoly game is more complex because firms must consider not only the behaviors of the consumers, but also the reactions of the competitors i.e. they form expectations concerning how

* Corresponding author. Tel.: +302531039819; fax: +302531039830. E-mail address: gsarafop@ierd.duth.gr

2212-5671 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review will be under responsibility of Department of Accountancy and Finance, Eastern Macedonia and Thrace Institute of Technology, Kavala, Greece.

doi: 10.1016/S2212-5671(15)00016-7

their rivals will act. Cournot, in 1838 has introduced the first formal theory of oligopoly. He treated the case with naive expectations, so that in every step each player (firm) assumes the last values that were taken by the competitors without estimation of their future reactions. In particular, we consider differentiated products and focus on the dynamic role played by the degree of horizontal product differentiation. While Cournot 1838 considered a duopoly with a single homogeneous product, more recently the economic literature offered duopoly models with (horizontal) differentiated products (see, e.g., Dixit, 1979; Singh and Vives, 1984) which allow goods and services to be substitutes or complements, in models with a standard linear demand structure.

Expectations play an important role in modelling economic phenomena. A producer can choose his expectations rules of many available techniques to adjust his production outputs. In this paper we study the dynamics of a duopoly model where each firm behaves with different expectations strategies. This kind of beliefs is common in real world problems such as economic, biology and social sciences problems. We consider a duopoly model where each player forms a different strategy in order to compute his expected output. We take firm 1 to represent a boundedly rational player while firm 2 has naive expectations. Each player adjusts his outputs towards the profit maximizing amount as target by using his expectations rule. Some authors considered duopolies with homogeneous expectations and found a variety of complex dynamics in their games, such as appearance of strange attractors (Agiza, 1999, Agiza et al., 2002, Agliari et al., 2005, 2006, Bischi, Kopel, 2001, Kopel, 1996, Puu,1998). Also models with heterogeneous agents were studied (Agiza, Elsadany , 2003, 2004, Agiza et al., 2002, Den Haan , 20013, Fanti, Gori, 2012, Tramontana , 2010, Zhang , 2007).

The main result of the present analysis is that the proposition of Fanti Gori, 2012, is not true if the cost function is quadratic. In this case there are trajectories of the system in which an increase in product differentiation may destabilize the unique Cournot-Nash equilibrium, but we show that in this case there are stable trajectories and a higher (lower) degree of product differentiation does not tend to destabilize (stabilize) the economy. Moreover, from a mathematical point of view, we show that the destabilization of the fixed point can occur through a flip bifurcation and also that a cascade of flip bifurcations may lead to periodic cycles and deterministic chaos.

The paper is organized as follows: In Section 2, the dynamics of a differentiated duopoly game with heterogeneous expectations, linear demand and quadratic cost functions is analyzed. The existence, local stability and bifurcation of the equilibrium points are also analyzed. In Section 3 numerical simulations are used to show complex dynamic via computing Lyapunov numbers, and sensitive dependence on initial conditions.

2. The game

2.1. Construction of the game

In oligopoly game players can choose simple expectation rules such as naive or complicated as adaptive expectations and bounded rationality. The players can use the same strategy (homogeneous expectations) or can use different strategy (heterogeneous expectations). In this study we consider heterogeneous players such that each player think with different strategy to maximize his output. Two different players expectations are proposed; bounded rational player and naive player. We consider a simple Cournot-type duopoly market where firms (players) produce differentiated goods and offer them at discrete-time periods t = 0, 1, 2, ... on a common market. At each period t, every firm must form an expectation of the rival's output in the next time period in order to determine the corresponding profit-maximizing quantities for period t+1. If x (resp. y) is the quantity of product of variety 1 (resp. 2), the inverse demand functions (as a function of quantities) are given by the following equations (Fanti, Gori, 2012)

px = a — x — dy, p.2 = a — y — dx,

We suppose that the cost functions are quadratic:

Cx{x) = cx2, C2(y) = cy-

where a, b , c >0, -1 < d < 1.

d represents the degree of horizontal product differentiation. More in detail, if d=0, then goods and services of

variety 1 and 2 are independent. This implies that each firm behaves as if it were a monopolist in its own market; if d=1, then products 1 and 2 are perfect substitutes or, alternatively, homogeneous; 0 < d <1 describes the case of imperfect substitutability between goods. The degree of substitutability increases, or equivalently, the extent of product differentiation decreases as the parameter d raises; a negative value of d instead implies that goods 1 and 2 are complements, while d = -1 reflects the case of perfect complementarity. With these assumptions the profits of the firms are given by

IIv(x,y) = x(a — x — dy) — ex2, U2(x,y) = y(a — x — dy) — cy2 (3)

Then the marginal profit at the point (x, y) of the strategy space is given by 3IL , N <9IL

-L = a- 2(1 + c)x - dy, -- = a- 2y(l + c) - dx (4)

—^ = 0&y = ——(a-dx) (5)

dy 2(1 + c)

We suppose that the first firm decides to increase its level of adaptation if it has a positive marginal profit, or decreases its level if the marginal profit is negative (bounded rational player) and the second is a naive player. If k > 0, the dynamical equations of the players are:

x(t + l)-x(t) = ^ y(t+1) = arg max JJ2 (x(t) , y(t)) (6)

X\t) UX y

The dynamical system of the players is described by

x(t + 1) = x(t) + kx(t)[a - 2(1 + c)x(t) - dy(t)\

2(1 + c)

We will focus on the dynamics of the system (6) to the parameter d .

2.2. The Equilibria of the game

The equilibria of the dynamical system (7) are obtained as nonnegative solutions of the algebraic system

kx(t)[a - 2(1 + c)x{t) - dy(t)} = 0

y(t) =---\a-dx(t) ] (8)

yyJ 2(1 + c) WJ

which obtained by setting x(t + 1) = x(t), y(t + 1) = y(t) in Eq. (7) and we can have two equilibriums

E0 = (0,a / 2(1 + c)) and E* = (x\y), where

x = y =

2(1 + c) + d

The fixed point E0, is the boundary equilibrium. The equilibrium E is called Nash equilibrium. The study of the

local stability of equilibrium solutions is based on the localization on the complex plane of the eigenvalues of the Jacobian matrix of the two dimensional map (Eq. (7)). In order to study the local stability of equilibrium points of the model (7), we consider the Jacobian matrix J(x, y) along the variable strategy (x, y)

J(x,y) =

1 + k-1 - 2kx(l + c) -kdx

2(1 + c)

The matrix

1 + k[a

J(Eq) =

2(1 + c)

2(1 + c)

has two eigenvalues:

\ = 0,A2 = 1 + ka

2(1 + c)

It follows thatÀ2 > 1, and then ^0is unstable fixed point for the system (7). The Nash equilibrium E is locally stable if the following conditions are hold (Gandolfo, 2010)

(i) 1 - D > 0 (it) 1 - T + D > 0 (Hi) 1 + T + D > 0

where T is the trace and D is the determinant of the Jacobian matrix

J{E*) =

1 - 2k(l + c)x -kdx

2(1 + c)

1-D = 1 +

kdd 2(1 + c) '

■x > 0

1 - T + D = —-[2(1 + c) + d] [2(1 + c) - d] > 0 (16)

2(1 + c)

The conditions (i) and (ii) of Eq. (13) are always satisfied and then the condition (iii) is the condition for the local stability of the Nash equilibrium. This condition becomes

-akd2 + 4(1 + c)d + 4(1 + c)2(2 - ak) > 0 (17)

The discriminant of Eq. (17) is positive if an only if

A = 16(1 + cf [—(ak)2 + 2ak + 1] > 0 ^ -(ak)2 + 2a/c + 1 > 0 ^ ak G (0,1 + ^2) (18)

It follows that Eq. (17) is verified if and only if

\ak G (0,1+ V2) Id G

[afc G (0,1 +>6) d G (dvd2)

3. Numerical Simulations

4(1 + c) -1 ± -y/—(aA;)2 + 2aA; + 1 ^ --- <20)

the two real roots of Eq. (17). It follows that:

Proposition. The Nash equilibrium of the dynamical system Eq. (7) is locally stable if and only if

We show that the model gives more complex chaotic and unpredictable trajectories as a consequence of change in the parameter of horizontal product differentiation and a higher (lower) degree of product differentiation (weaker or fiercer competition) destabilize (stabilize) the economy. To provide some numerical evidence for the chaotic behavior of the system Eq. (7), as a consequence of change in the parameter d we present various numerical results here to show the chaoticity, including its bifurcations diagrams, strange attractor, Lyapunov numbers and sensitive dependence on initial conditions (Kulenovic, M., Merino, O., 2002). In order to study the local stability properties of the equilibrium points, it is convenient to take

a = 10, k = 0.2 ^ = 0, d2 = 2(1 + c) > 1 (22)

Numerical experiments are computed to show the bifurcation diagram with respect to d, strange attractor of the system Eq.(7) in the phase plane (x,y) and the Lyapunov numbers . Fig. 1 show the bifurcation diagrams with respect to the parameter d. In this figure one observes complex dynamic behavior such as cycles of higher order and

chaos. Fig. 2 show the graph of strange attractor and Lyapunov numbers of the orbit of (0.1, 0.1) for k = -0.98. From these results when all parameters are fixed and only d is varied the structure of the game becomes complicated through period doubling bifurcations, more complex bounded attractors are created which are aperiodic cycles of higher order or chaotic attractors.

To demonstrate the sensitivity to initial conditions of the system Eq.(7), we compute two orbits with initial points (0.1, 0.1) and (0.101, 0.1), respectively. Fig. 3 shows sensitive dependence on initial conditions for x-coordinate of the two orbits, for the system Eq.(7), plotted against the time with the parameters values a = 10, k = 0.2, d = -0.98. At the beginning the time series are indistinguishable; but after a number of iterations, the difference between them builds up rapidly. From Fig. 3 we show that the time series of the system Eq. (7) is sensitive dependence to initial conditions, i.e. complex dynamics behaviors occur in this model.

We observe that a higher degree of product market differentiation may destabilize the unique Cournot-Nash equilibrium, while also showing the existence of deterministic chaos. This result suggests a twofold effect: while an increase in the extent of product differentiation tends to increase profits, it may also cause the loss of stability of the equilibrium through a flip bifurcation.

a = 10, k = 0.1, c = l^=4(l-^)<-l, d2 = 4(1 + S) > 1

From Eq. (21) for each d in the interval (- 1, 1) the Nash equilibrium E* = , is locally

[4 + d 4 +

asymptotically stable (Fig. 4). Therefore, in the case of quadratic costs there are stable trajectories and a higher (lower) degree of product differentiation does not tend to destabilize (stabilize) the market.

Fig. 1. Bifurcation diagrams with respect to the parameter d against variable x (left) and against y (right) for a =10, k= 0.2, with 550 iterations of

the map Eq. (7).

Fig.2. Phase portrait (strange attractor) with 2000 iterations of the map (left) and Lyapunov numbers versus the number of iterations (right), of the

orbit, orb. (0.1, 0.1), for a =10, k = 0.2, d = - 0.98

Fig.3. Sensitive dependence on initial conditions, for x-coordinate plotted against the time: The two orbits orb.(0.1, 0.1) (left) and orb.(0.1, 0.101) (right), for the system (6), with the parameters values a = 10, k = 0.2, d = -0.98.

Fig.4. Bifurcation diagrams with respect to the parameter d against variable x (left) and against y (right) for a =10, k= 0.1, with 550 iterations of

the map Eq.(7).

4. Conclusion

We show that the parameter of horizontal product differentiation (for some values of the others parameters) may change the stability of equilibrium and cause a structure to behave chaotically. For some values of this parameter there is a stable Nash equilibrium. Decreasing these values, the equilibrium becomes unstable, through period-doubling bifurcation. Also, we show that in our case of quadratic costs there are stable trajectories for each d in the interval (-1, 1), and a higher (lower) degree of product differentiation does not tend to destabilize the market.

Acknowledgements

The author thanks the two anonymous referees for interesting remarks and useful suggestions. References

Agiza, H.N., 1999. On the analysis of stability, bifurcation, chaos and chaos control of Kopel map. Chaos, Solitons & Fractals 10, 1909-1916.

Agiza, H.N., Hegazi, A.S., Elsadany, A.A., 2002. Complex dynamics and synchronization of duopoly game with bounded rationality. Mathematics and Computers in Simulation 58, 133-146.

Agiza, H.N., Elsadany, A.A., 2003. Nonlinear dynamics in the Cournot duopoly game with heterogeneous players. PhysicaA 320, 512-524.

Agiza, H.N., Elsadany, A.A., 2004. Chaotic dynamics in nonlinear duopoly game with heterogeneous players. Applied Mathematics and Computation 149, 843-860.

Agiza HN, Hegazi AS, Elsadany AA. Complex dynamics and synchronization of duopoly game with bounded rationality. Math Comput Simulat.2002; 58: 133-46.

Agliari, A., Gardini, L., Puu, T., 2005. Some global bifurcations related to the appearance of closed invariant curves. Mathematics and Computers in Simulation 68, 201-219.

Agliari, A., Gardini, L., Puu, T., 2006. Global bifurcations in duopoly when the Cournot point is destabilized via a subcritical Neimark bifurcation. International Game Theory Review 8, 1-20.

Bischi, G.I., Kopel, M., 2001. Equilibrium selection in a nonlinear duopoly game with adaptive expectations. Journal of Economic Behavior & Organization 46, 73-100.

Bischi, G.I., Naimzada, A., 1999, Global analysis of a dynamic duopoly game with bounded rationality. In: Filar JA, Gaitsgory V, Mizukami K, (eds). (2000). Advances in dynamic games and applications, vol. 5. Basel: Birkhauser: 361-385.

Cournot, A., 1838. Recherches sur les Principes Mathématiques de la Théorie des Richessess. Hachette, Paris.

Day, R., 1994. Complex Economic Dynamics. MIT Press, Cambridge.

Den Haan,W.J., 2001. The importance of the number of different agents in a heterogeneous asset-pricing model. Journal of Economic Dynamics and Control 25, 721-746.

Dixit, A.K., 1979. A model of duopoly suggesting a theory of entry barriers. Bell Journal of Economics 10, 20-32.

Fanti, L., Gori., 20012 The dynamics of a differentiated duopoly with quantity competition, Economic Modelling, 29, 421-427

Gandolfo, G., 2010. Economic Dynamics, Forth ed. Springer, Heidelberg.

Kopel, M., 1996. Simple and complex adjustment dynamics in Cournot duopoly models. Chaos, Solitons & Fractals 7, 2031-2048

Kulenovic, M., Merino, O., 2002. Discrete Dynamical Systems and Difference Equations with Mathematica. Chapman & Hall/Crc.

Puu T. 1998, The chaotic duopolists revisited. J Econom. Behav. Org. 37: 385-94.

Singh, N., Vives, X., 1984. Price and quantity competition in a differentiated duopoly. The RAND Journal of Economics 15, 546-554.

Tramontana, F., 2010. Heterogeneous duopolywith isoelastic demand function. Economic Modelling 27, 350-357.

Wu, W., Chen, Z., Ip, W.H., 2010. Complex nonlinear dynamics and controlling chaos in a Cournot duopoly economic model. Nonlinear Analysis: Real World Applications 11, 4363-4377.

Zhang, J., Da, Q.,Wang, Y., 2007. Analysis of nonlinear duopoly game with heterogeneous players. Economic Modelling 24, 138-148.