Controls of the complex dynamics of a multi-market Cournot model
E. Ahmed
a
, M.F. Elettreby
a,b
a
Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
b
Mathematics Department, Faculty of Science, King Khaled University, Abha 9004, Saudi Arabia
abstract article info
Article history:
Accepted 12 November 2013
Keywords:
Cournot
Bounded rationality
Steady state
Stability
Bifurcation
Chaos
In this paper, a multi-market Cournot game is proposed based on a specific inverse demand function. The game is
studied statically and dynamically. Puu's incomplete information approach, as a realistic method, is used to con-
tract the corresponding dynamical model under this function. Therefore, some stability analysis is carried out on
the model to detect the stability and instability conditions of the system's Nash equilibrium. Based on that anal-
ysis some dynamic phenomena such as bifurcation and chaos are found. Under certain assumption, chaos control
is performed in order to control the monopolistic model. Furthermore, a dynamic multi-market Cournot model is
introduced.
© 2013 Elsevier B.V. All rights reserved.
1. Introduction
Cournot competition is an economic model used to describe the
competition between some companies on the amount of output they
will produce (Cournot, 1897). Here, a generalization of this game to
the case of two markets is done. It is shown that the resulting dynamics
is quite rich. In Sections 2 and 3, we begin with monopoly in two mar-
kets using Puu's incomplete information dynamics (Ahmed et al.,
2006). It is more realistic than the standard bounded rationality since
profits may not be known as functions but only as quantities (Puu,
1991). The dynamic behavior shows cycles and chaos. In Section 4,
chaos control is studied for the monopoly case. In Section 5, duopoly
case is studied in two markets. The dynamical features of the proposed
system are studied numerically. Due to the difficulty in dealing with
such systems, because of complex phenomena found such as bifurcation
and chaos, some numerical simulation is carried out to investigate the
impact of those phenomena on the behavior of the system.
2. Static multi-market Cournot model
In this model, the products are near substitutes but different in the
quality levels. So, firms can charge different prices for different markets.
Suppose we have n firms (n ≥ 2) that compete in two inter-related
markets A and B. Bowley (1924) has introduced the demand function
in the form;
p
i
¼ α
i
-q
i
-θ
X
j≠i
q
j
; ð1Þ
where 0 ≤ θ ≤ 1. A lot of papers (Chakrabarti and Haller, 2007; Dixit,
1979; Spence, 1976) used this form in their work. Billand et al. (2010)
used the same demand function with θ = 1. Let the demand in the
market A for the firm i be;
p
i
¼ a
i
-q
i
-
X
j≠i
q
j
; ð2Þ
where p
i
is the firm i's price, q
i
is the firm i's quantity and the constant
a
i
N 0 for market A. The demand in the market B for the firm i is;
P
i
¼ b
i
-Q
i
-
X
j≠i
Q
j
; ð3Þ
where P
i
is the firm i's price, Q
i
is the firm i's quantity and the constant
b
i
N 0 for market B.
The cost function of the firm i is;
C
i
q
i
; Q
i
ð Þ¼
1
2
q
i
þ Q
i
ð Þ
2
; ð4Þ
hence the profit of the firm i is given by;
π
i
¼ p
i
q
i
þ P
i
Q
i
-C
i
q
i
; Q
i
ð Þ: ð5Þ
This form of the profit function (5) satisfies the joint disecon-
omies
∂
2
π
i
∂ q
i
∂ Q
i
¼ -1b0
and strategic substitute
∂
2
π
i
∂ Q
i
∂ Q
i
¼ -1b0
properties.
Economic Modelling 37 (2014) 251–254
E-mail address: mohfathy@mans.edu.eg (M.F. Elettreby).
0264-9993/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.econmod.2013.11.016
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