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Procedia Economics and Finance 33 (2015) 358 - 366

7th International Conference, The Economies of Balkan and Eastern Europe Countries in the changed world, EBEEC 2015, May 8-10, 2015

Complexity in a duopoly game with homogeneous players, convex, log-linear demand and quadratic cost functions

Georges Sarafopoulos *

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

Abstract

In this study we investigate the dynamics of a nonlinear discrete-time duopoly game, where the players have homogeneous expectations. We suppose that the cost function is quadratic and the demand is a convex and log- linear function. 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 speed of adjustment of the players. If this parameter is varied, the stability of Nash equilibrium is lost through period doubling bifurcations. The chaotic features are justified numerically via computing Lyapunov numbers and sensitive dependence on initial conditions.

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

Peer-reviewunder responsibility of Department of Accountancy and Finance, Eastern Macedonia and Thrace Institute of Technology Keywords: Cournot duopoly game; Discrete dynamical system; Homogeneous expectations; 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 their rivals will act. Cournot, in 1838 has introduced the first formal theory of oligopoly. He treated the case with

* 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 under responsibility of Department of Accountancy and Finance, Eastern Macedonia and Thrace Institute of Technology doi: 10.1016/S2212-5671(15)01720-7

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.

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 homogeneous expectations strategies. We consider a duopoly model where each player forms a strategy in order to compute his expected output. 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).

In the real market producers do not know the entire demand function, though it is possible that they have a perfect knowledge of technology, represented by the cost function. Hence, it is more likely that firms employ some local estimate of the demand. This issue has been previously analyzed by Baumol and Quandt, 1964, Puu 1995, Naimzada and Ricchiuti, 2008, Askar, 2013, Askar, 2014. Efforts have been made to model bounded rationality to different economic areas: oligopoly games (Agiza, Elsadany, 2003, Bischi et al, 2007); financial markets (Hommes, 2006); macroeconomic models such as multiplier-accelerator framework (Westerhoff,2006). In particular, difference equations have been employed extensively to represent these economic phenomenons (Elaydi, 2005; Sedaghat, 2003). Bounded rational players (firms) update their production strategies based on discrete time periods and by using a local estimate of the marginal profit. With such local adjustment mechanism, the players are not requested to have a complete knowledge of the demand and the cost functions (Agiza, Elsadany, 2004, Naimzada, Sbragia, 2006, Zhang et al, 2007, Askar, 2014). All they need to know is if the market responses to small production changes by an estimate of the marginal profit.

The present study extends Askar 2014. We investigate the dynamics of a nonlinear discrete-time duopoly game, where the players are boundedly rational and they have homogeneous expectations. We suppose that the cost function is quadratic and the demand function is convex and log- linear. 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 speed of adjustment of the players. 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 the duopoly game with homogeneous expectations, convex and log-linear demand and quadratic cost functions is analyzed. The existence and local stability of the equilibrium points are also analyzed. In Section 3 numerical simulations are used to show complex dynamics via computing Lyapunov numbers, and sensitive dependence on initial conditions.

2. The game

In oligopoly game players can choose simple expectation rules such as naïve 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 homogeneous players such that each player thinks with the same strategy to maximize his output (bounded rational player). Let us consider a market where two firms produce homogeneous commodities. Production decisions are taken at discrete time periods t = 0,1,2,... We consider a simple Cournot-type duopoly market where firms (players) produce the same good and offer them at discrete-time periods 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 the player 1 (resp. 2), the inverse demand functions (as a function of quantities) are given by the following equation

p = a — b ln(x + y )

The direct demand function is exponential

q = x + y = e b (2)

Thus it is convex and log-linear in the price. A function is log-linear if and only if its logarithm is linear. The marketing interpretation of log-linearity is that the rate of growth of the quantity is constant. We suppose that the cost functions of the players are quadratic:

C1(x ) = cx2, C2(y) = cy2 (3)

where a,b,c > 0

With these assumptions the profits of the firms are given by

Hj(x, y) = x[a - b ln(x + y)] - cx2, n2(x, y) = y[a - b ln(x + y)] - cy2 (4)

Then the marginal profit at the point (x, y) of the strategy space is given by

^^ b ^^ b -1 = a - bln(x + y) - x(2c H--), -2 = a - bln(x + y) - y(2c H--) (5)

dx x + y dy x + y

We suppose that each 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). If k > 0 the dynamical equations of the players are:

x(t + 1) - x(t) _ <9^ ' — k "

y(t + 1) - y(t) = k d^2

k is the speed of adjustment of the players, it is a positive parameter which gives the extent of production variation of the firm following a given profit signal. Moreover it captures the fact that relative effort variations are proportional to the marginal profit.

The dynamical system of the players is described by

x(t + 1) = x(t) + kx(t)[a - b ln (x + y) - x(2c + • y(t + 1) = y(t) + ky(t)[a - b ln(x + y) - y(2c +

x + y f-)

We will focus on the dynamics of this system to the parameter k . 2.1. The Equilibria of the game

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

kx[a — b ln(x + y) — x(2c H--b— )]

ky[a — b ln(x + y) — y(2c H--)]

which obtained by setting x(t + 1) = x(t), y(t + 1) = y(t) in Eq. (8). We can have two boundary equilibria E1 = (0, y1), where y1 is the unique solution of the equation

a - b - 2cy - b ln y = 0 (9)

E2 = (x2,0), where x2 is the unique solution of the equation

a - b - 2cx - b ln x = 0 (10)

in the interval (0, +rc>), and the Nash equilibrium E = (x , y ) which is the solution of the system:

a — b ln(x + y) — x(2c + ■

-) = 0

a - b ln(x + y) - y(2 c H--b— ) = 0

Then, x = y is the unique solution in the interval (0, +<x>) of the equation:

a - b ln(2x) - 2cx - 2 = 0 (12)

The study of the local stability of the equilibrium 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) =

f (x, y) = x + kx-1 = x + kx[a — b ln(x + y) — x(2c H--b—)]

dx x + y

g(x, y) = y + ky ^^ = y + ky[a - b ln(x + y) - y(2c H--b—)]

ay x + y

The Jacobian matrix

J E) =

fx E1) 0

9X E) gv E)

has two eigenvalues:

\ = fx E1)-^ = gy E)

It follows that A1 = 1 + k(b + 2cy1) > 1, and then E1 is unstable fixed point for the system (Eq.(7)). The matrix

J (E2) =

fx (E2 ) fy E)

o gv (E2) _

has two eigenvalues:

Aj = fx (E2)-A2 = gv (E2)

A2 = 1 + k(b + 2cx2) > 1, and then E2 is also an unstable fixed point for the system Eq. (7). Since

fx(E*) = gy(E*) = 1 - kx*(2c + -3*)

f (E *) = gx (E *) = kx

= .ki dx dy 4

The Jacobian matrix

J (E* ) =

X (E * ) fy (E * ) gx (E * ) gy (E * )

X E ) fy E )

fy (E * ) X (E * ) _

*d2 n,

1 + kx 1

*d2 n,

kx- 92 ni

*d2 n,

1 + kx 1

has two real eigenvalues

A1 = I (E* ) + fy (E* ) = 1 - k(2cx* + b) \ = X (E* ) - fv (E* ) = 1 - k (2cx* + b)

Then, the Nash equilibrium E is locally stable if the following conditions are hold:

N < 1' N < 1

Therefore, it is locally asymptotically stable under the condition

0 < k <

2cx + b It follows that:

Proposition. The Nash equilibrium E(x , x ) of the dynamical system Eq. (7) is locally asymptotically stable if

0 < k <

2cx + b

3. Numerical Simulations

To provide some numerical evidence for the chaotic behavior of the system Eq. (7), as a consequence of change in the parameter k of the speed of adjustment, 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 the parameters values as follows a = 2, b = 2, c = 1. Numerical experiments are computed to show the bifurcation diagram with respect to k , strange attractor of the system Eq. (7) in the phase plane (x, y), and the Lyapunov numbers. Fig. 1 shows the bifurcation diagrams with respect to the parameter k against variable x and y i.e. the orbit of the point A (0.1, 0.2) with 850 iteration of the map Eq. (7) and Lyapunov numbers of the same orbit for a = 2, b = 2, c = 1. If the Lyapunov number is greater of 1, one has evidence for chaos.

In these figures the Nash equilibrium E* (0.5, 0.5) is locally asymptotically stable for 0 < k < 2/3 . For k > 2/3 the Nash equilibrium E* becomes unstable, and one observes complex dynamic behavior such as cycles of higher order and chaos. Fig. 2 shows the graphs of the orbit of the point (0.1, 0.2) (strange attractors) for a = 2, b = 2, c = 1, k = 0.82 and a = 2, b = 2, c = 1, k = 0.85 . From these results when all parameters are fixed and only k 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, 0.2) and (0.1001, 0.2), 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 = 2, b = 2, c = 1, k = 0.85. 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.

O.B [-1-1-1-1-1-1-1-1-1-i-1-1-1-1-1-1-1.40 prf

Fig.1. Bifurcation diagram with respect to the parameter k against variable x and y, with 850 iterations of the map Eq. (7) (left) and Lyapunov numbers of the orbit of the point A(0.1, 0.2) versus the number of iterations (right), for a =2, b=2, c = 1, k = 0.85

Fig.2. Phase portrait (strange attractors). The orbit of (0.1, 0.2) with 3000 iterations of the map Eq. (7) for a = 2, b = 2, c = 1, k = 0.82, with box dimension approximately equal to 1.14 (left) and for a = 2, b = 2, c = 1, k = 0.85, with box dimension approximately equal to 1.15 (right).

0 20 +3 63 Si 100 0 20 40 63 Si 100

Fig.3. Sensitive dependence on initial conditions, for x-coordinate plotted against the time: The two orbits: the orbit of (0.1, 0.2) (left) and the orbit of (0.1001, 0.2) (right), for the system Eq. (7), with the parameters values a = 2, b = 2, c = 1, k = 0.85.

4. Conclusion

In this paper, we analyzed through a discrete dynamical system based on the marginal profits of the players, the dynamics of a nonlinear discrete-time duopoly game, where the players have homogeneous expectations. We suppose that the cost function is quadratic and the demand function is convex and log- linear. The stability of equilibria, bifurcation and chaotic behavior are investigated. We show that a parameter (speed of adjustment) may change the stability of equilibrium and cause a structure to behave chaotically. For low values of this parameter there is a stable Nash equilibrium. Increasing these values, the equilibrium becomes unstable, through period-doubling bifurcation.

Acknowledgements

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

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