Complicated dynamics of an interacting economic model
We study a N-dimensional dynamical system that presents interacting economic models, the so-called quantity setting oligopoly. Parameter regions of stability of the Nash equilibrium are obtained and the phase diagram for stability of the synchronized motion are investigated. We also consider the...
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irk-123456789-1770092021-02-10T01:25:50Z Complicated dynamics of an interacting economic model Stepanenko, O. Maistrenko, Y.L. Yousefi, S. We study a N-dimensional dynamical system that presents interacting economic models, the so-called quantity setting oligopoly. Parameter regions of stability of the Nash equilibrium are obtained and the phase diagram for stability of the synchronized motion are investigated. We also consider the problem of cluster formation. In particular, the 2-cluster state is studied extensively. A locally stable Nash equilibrium and quasiperiodic motions are present in this state. The regions of stability and the bifurcations of stable trajectories are investigated and corresponding bifurcation diagram are obtained Вивчається N-вимiрна динамiчна система, яка задає взаємодiючi економiчнi моделi i є так званою кiлькiсною визначаючою олiгополлю. Отримано областi параметрiв стiйкостi рiвноважного стану Неша та вивчено фазову дiаграму стiйкостi синхронного руху. Також розглянуто проблему утворення кластерiв, зокрема, детально вивчено стан з двома кластерами. Цей стан має локально стiйкий рiвноважний стан Неша, а також має мiсце квазiперiодичний рух. Дослiджено областi стiйкостi i бiфуркацiї стiйких траєкторiй i отримано вiдповiдну бiфуркацiйну дiаграму. 2004 Article Complicated dynamics of an interacting economic model / O. Stepanenko, Y.L. Maistrenko, S. Yousefi // Нелінійні коливання. — 2004. — Т. 7, № 2. — С. 255-262. — Бібліогр.: 5 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177009 517.9 en Нелінійні коливання Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
We study a N-dimensional dynamical system that presents interacting economic models, the so-called
quantity setting oligopoly. Parameter regions of stability of the Nash equilibrium are obtained and the
phase diagram for stability of the synchronized motion are investigated.
We also consider the problem of cluster formation. In particular, the 2-cluster state is studied extensively. A locally stable Nash equilibrium and quasiperiodic motions are present in this state. The regions of
stability and the bifurcations of stable trajectories are investigated and corresponding bifurcation diagram
are obtained |
| format |
Article |
| author |
Stepanenko, O. Maistrenko, Y.L. Yousefi, S. |
| spellingShingle |
Stepanenko, O. Maistrenko, Y.L. Yousefi, S. Complicated dynamics of an interacting economic model Нелінійні коливання |
| author_facet |
Stepanenko, O. Maistrenko, Y.L. Yousefi, S. |
| author_sort |
Stepanenko, O. |
| title |
Complicated dynamics of an interacting economic model |
| title_short |
Complicated dynamics of an interacting economic model |
| title_full |
Complicated dynamics of an interacting economic model |
| title_fullStr |
Complicated dynamics of an interacting economic model |
| title_full_unstemmed |
Complicated dynamics of an interacting economic model |
| title_sort |
complicated dynamics of an interacting economic model |
| publisher |
Інститут математики НАН України |
| publishDate |
2004 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/177009 |
| citation_txt |
Complicated dynamics of an interacting economic model / O. Stepanenko, Y.L. Maistrenko, S. Yousefi // Нелінійні коливання. — 2004. — Т. 7, № 2. — С. 255-262. — Бібліогр.: 5 назв. — англ. |
| series |
Нелінійні коливання |
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AT stepanenkoo complicateddynamicsofaninteractingeconomicmodel AT maistrenkoyl complicateddynamicsofaninteractingeconomicmodel AT yousefis complicateddynamicsofaninteractingeconomicmodel |
| first_indexed |
2025-07-15T14:58:06Z |
| last_indexed |
2025-07-15T14:58:06Z |
| _version_ |
1837725371501379584 |
| fulltext |
UDC 517.9
COMPLICATED DYNAMICS
OF AN INTERACTING ECONOMIC MODEL
СКЛАДНА ДИНАМIКА
ВЗАЄМОДIЮЧОЇ ЕКОНОМIЧНОЇ МОДЕЛI
O. Stepanenko, Yu. Maistrenko
Inst. Math. Nat. Acad. Sci. Ukraine
Tereshchenkivs’ka Str., 3, Kyiv, 01601, Ukraine
Sh. Yousefi
SDU-Univ. Southern, Denmark
We study a N -dimensional dynamical system that presents interacting economic models, the so-called
quantity setting oligopoly. Parameter regions of stability of the Nash equilibrium are obtained and the
phase diagram for stability of the synchronized motion are investigated.
We also consider the problem of cluster formation. In particular, the 2-cluster state is studied extensi-
vely. A locally stable Nash equilibrium and quasiperiodic motions are present in this state. The regions of
stability and the bifurcations of stable trajectories are investigated and corresponding bifurcation diagram
are obtained.
Вивчається N -вимiрна динамiчна система, яка задає взаємодiючi економiчнi моделi i є так зва-
ною кiлькiсною визначаючою олiгополлю. Отримано областi параметрiв стiйкостi рiвноваж-
ного стану Неша та вивчено фазову дiаграму стiйкостi синхронного руху. Також розглянуто
проблему утворення кластерiв, зокрема, детально вивчено стан з двома кластерами. Цей стан
має локально стiйкий рiвноважний стан Неша, а також має мiсце квазiперiодичний рух. Дослiд-
жено областi стiйкостi i бiфуркацiї стiйких траєкторiй i отримано вiдповiдну бiфуркацiйну
дiаграму.
1. Introduction. Model. The Cournot model of imperfect competition between economic agents
is a well-known concept in mathematical economics [1 – 3]. This model elaborates on a parti-
cular scenario when the market structure departs from monopoly towards a competitive frame-
work. The simple case of the Cournot duopoly involves two agents competing for the lion share
of the market. The more complicated case of the Cournot oligopoly involves an increasing
number of competing agents who are able to collectively exert control over the supply of a
given commodity. In all these cases, it is assumed that the expectations on the competitor’s acti-
ons can be used to form the agent’s own preferred action, which is profit-maximizing choice
given by these expectations.
In the present study, a simple framework of the models previously studied in [4] and [5] is
extended to the general case of oligopoly competition among a large number of agents in the
market. A quantity-setting oligopoly where the agents provide a homogeneous good is consi-
dered. Despite the fairly restrictive assumption of the model, the obtained results appear to
be useful in understanding the underlying dynamical properties of the general model. In doing
so, the focus has been directed towards the mathematical description of the model followed by
elaborations on possible economic interpretations of the obtained results.
c© O. Stepanenko, Yu. Maistrenko, and Sh. Yousefi, 2004
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2 255
256 O. STEPANENKO, YU. MAISTRENKO, AND SH. YOUSEFI
The time evolution of a dynamic oligopoly game with N competing firms is modelled by
a discrete dynamical system obtained by the iteration of a N -dimensional noninvertible map.
Output quantity of each firm is adjusted in response to the quantities observed in the previous
period. Making the same assumptions as introduced in [5] and taking reaction functions in a
form
ri = µi
∑
j 6=i
xj(1− xj), i = 1, . . . , N,
we have oligopoly game modelled by the map
x
(n+1)
i = (1− λ)x(n)
i + λµ
∑
j 6=i
f(x(n)
j ), (1)
where x(n)
i > 0, i = 1, . . . , N , is a quantity of good produced by the i-th firm at the n-th period
of time, λ ∈ [0; 1] represent the speed of the adjustment process for achieving optimal quantity
producing, µi > 0 represent an extent of the interfirm externality incorporated in cost functions
of the firms, f is a quadratic map f(x) = x(1− x).
Let’s denote
F :
x
(n)
1
x
(n)
2
...
x
(n)
N
→
x
(n+1)
1
x
(n+1)
2
...
x
(n+1)
N
.
The Jacobian matrix of map (1) is
DF =
1− λ λµf ′(x2) . . . λµf ′(xN )
λµf ′(x1) 1− λ . . . λµf ′(xN )
...
...
. . .
...
λµf ′(x1) λµf ′(x2) . . . 1− λ
.
Below in this study we assume xi > 0, i = 1, . . . , N .
2. Synchronized state. Concider a synchronized state, i. e., when dynamics of the map (1) is
confined within the diagonal
D = {x1 = x2 = . . . = xN ≡ x}
and the map can be written in the form
g : x → (1− λ)x+ (N − 1)λµf(x). (2)
For the oligopoly modelled, it means that all the firms produce the same quantity of goods.
Map (2) depends on two parameters λ, µ and can be reduced to the logistic map f̃ = ãx̃(1− x̃),
where
ã =
(1− λ+ (N − 1)λµ)2
(N − 1)λµ
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COMPLICATED DYNAMICS OF AN INTERACTING ECONOMIC MODEL 257
by substitution
x =
(1− λ+ (N − 1)λµ)
(N − 1)λµ
x̃.
Proposition 1. Eigenvalues of the Jacobian matrix DF of the map (1) in the diagonal D are
equal to
ν1 = 1− λ+ (N − 1)λµf ′(x),
ν2,...,N = 1− λ− λµf ′(x).
(3)
Proof. The characteristic equation for eigenvalues of the Jacobian matrix DF in the di-
agonal D assumes the form
|DF |D − νI| =
∣∣∣∣∣∣∣∣∣
1− λ− ν λµf ′(x) . . . λµf ′(x)
λµf ′(x) 1− λ− ν . . . λµf ′(x)
...
...
. . .
...
λµf ′(x) λµf ′(x) . . . 1− λ− ν
∣∣∣∣∣∣∣∣∣ = 0.
By matrix transformation, it is reduced to the equation
(1− λ− ν + (N − 1)λµf ′(x))(1− λ− ν − λµf ′(x))(N−1) = 0,
which gives us solution (3).
Remark 1. For further consideration let’s denote ν1 =: ν‖, ν2,...,N =: ν⊥, since ν1 corres-
ponds to the eigenvector which is parallel to the diagonal and ν2,...,N corresponds to eigenvectors
which are perpendicular to it.
2.1. Region of stability of the Nash equilibrium. Consider a nontrivial fixed point x∗ of the
map which is situated on the diagonal D (1) (It is the so-called symmetric Nash equilibria for
the oligopoly.) We can find it by solving the equation
(1− λ)x+ (N − 1)λµx(1− x) = x.
Then
x∗ = 1− 1
(N − 1)µ
.
Eigenvalues of the Jacobi matrix in the fixed point are equal to
ν‖|x=x∗ = 1 + λ− (N − 1)λµ,
ν⊥|x=x∗ = 1− λ− λ
(
2
N − 1
− µ
)
.
The region of stability of x∗ in the parameters space (λ, µ) is bounded by the curves
µ1 =
1
N − 1
, [ν‖ = 1], λ2 =
2
(N − 1)µ− 1
, [ν‖ = −1], µ3 = 1 +
2
N − 1
, [ν⊥ = 1].
The region is represented in Fig. 1 (marked as P (s)
1 ) for different N .
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
258 O. STEPANENKO, YU. MAISTRENKO, AND SH. YOUSEFI
Fig. 1. Bifurcation diagram of the behavior on the diagonal for N = 10 (a), N = 50 (b), N =
= 100 (c). P (s)
0 represents the region of stability of the trivial fixed point (0, . . . , 0); P (s)
1
represents the region of stability of a fixed point in the diagonal; P (s)
2 represents the regi-
on of stability of period 2 orbit; P (s)
4 represents the region of stability of period 4 orbit;
Sothers represents the region where the trajectories in the diagonal of the period > 4
and those that have chaotic motion are transversally stable.
2.2. Bifurcation scenario in the diagonal. Regions of stability of period 2 and period 4 cycles
and some other cycles which lie on the diagonal are shown in Fig. 1 ( denoted by P (s)
2 , P (s)
4 ,
Sothers).
On the line µ1 =
1
N − 1
, a transcritical bifurcation occurs when the trivial fixed point
x0 = 0 loses its stability, and the nontrivial stable fixed point x∗ = 1 − 1
(N − 1)µ
, appears.
On the curve λ2 =
2
(N − 1)µ− 1
, there occurs a supercritical period doubling bifurcation (the
eigenvalue ν‖ = −1). The fixed point x∗ = 1− 1
(N − 1)µ
loses its stability and a stable period
2 cycle appears.
As λ and µ increase, new stability regions in the diagonal appear. The order of appearance
of these regions corresponds to periodic windows of the logistic map (on the line λ = 1 the
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
COMPLICATED DYNAMICS OF AN INTERACTING ECONOMIC MODEL 259
Fig. 2. A more detailed picture of the bifurcation diagram
on the diagonal for N = 10. It shows a period doub-
ling order of the appearance of stable trajectories
with a synchronized periodic motion.
Fig. 3. Regions of stability of a period 2 cycle in the diagonal,
with N = 10, 50, and 100, respectively.
map g completely coincides with the canonical form of the logistic map ax(1−x)). Fig. 2 shows
a more deteiled picture of the bifurcation diagram on the diagonal for N = 10.
Dependence of the regions’ sizes on the system size N is represented in Fig. 3. Thus, the
regions are strongly decreasing as N increases.
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
260 O. STEPANENKO, YU. MAISTRENKO, AND SH. YOUSEFI
3. Stability of the 2-cluster state. Consider a 2-cluster state that arises when all coordinates
xi are divided into two groups (clusters) with the dynamics being identical in each group. Thus
the total dynamics is confined to the 2-dimension manifold
D2 :
{
x1 = x2 = . . . = xN1 ,
xN1+1 = . . . = xN ,
where N1 is the number of elements in the first group. Denote N2 := N − N1, x := x1 = . . .
. . . = xN1 , y := xN1+1 = . . . = xN . Then, the dynamics of the map is described by the
2-dimension map
G2 :
(
x
y
)
→
(
(1− λ)x+ λµ(N1 − 1)f(x) + λµN2f(y)
(1− λ)y + λµN1f(x) + λµ(N2 − 1)f(y)
)
. (4)
To investigate the behaviour of map (1) in the manifold D2, first, “in-cluster” eigenvalues
ν1,2 should be found. We call them “in-cluster” since they describe the internal dynamics of
the 2-clusters developed in the manifold D2, that is, the behaviour of groups of firms rather
than individual firms. So, we solve the characteristic equation
|DG2 − νI| =
∣∣∣∣ 1− λ− ν + λµ(N1 − 1)f ′(x) λµN2f
′(y)
λµN1f
′(x) 1− λ− ν + λµ(N2 − 1)f ′(x)
∣∣∣∣ = 0.
The eigenvalues are found to be
ν1,2 = 1− λ+
λµ
2
(
(N1 − 1)f ′(x) + (N2 − 1)f ′(y)±
±
√
((N1 − 1)f ′(x)− (N2 − 1)f ′(y))2 + 4N1N2f ′(x)f ′(y)
)
. (5)
To evaluate the transversal stability of the 2-cluster state in the whole N -dimensional space, we
need to find all other transversal eigenvalues. They are found from the characteristic equation
of the form
|DF |D2 − νI| =
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
1− λ− ν λµf ′(x) . . . λµf ′(x) λµf ′(y) . . . λµf ′(y)
λµf ′(x) 1− λ− ν . . . λµf ′(x) λµf ′(y) . . . λµf ′(y)
...
...
. . .
...
...
...
...
λµf ′(x) λµf ′(x) . . . 1− λ− ν λµf ′(y) . . . λµf ′(y)
λµf ′(x) λµf ′(x) . . . λµf ′(x) 1− λ− ν . . . λµf ′(y)
...
...
...
...
...
. . .
...
λµf ′(x) λµf ′(x) . . . λµf ′(x) λµf ′(y) . . . 1− λ− ν
∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣
= 0.
By using matrix transformations, it can be reduced to the form∣∣∣∣∣ 1− λ− ν + (N1 − 1)λµf ′(x) N2λµf
′(y)
N1λµf
′(x) 1− λ− ν + (N2 − 1)λµf ′(y)
∣∣∣∣∣×
× (1− λ− ν − λµf ′(x))N1−1(1− λ− ν − λµf ′(y))N2−1 = 0.
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
COMPLICATED DYNAMICS OF AN INTERACTING ECONOMIC MODEL 261
Fig. 4. Stability regions of the fixed points (labeled by C(1:N2)
f ) and quasiperiodic attractors
(labeled by C(1:N2)
ch ) if the cluster ratio N1 : N2 is 1 : 9 (a), 1 : 49 (b), 1 : 99 (c).
Thus, there are two different transversal eigenvalues:
ν⊥,1 := ν3,...,N1+1 = 1− λ− λµf ′(x),
ν⊥,2 := νN1+2,...,N = 1− λ− λµf ′(y).
(6)
These eigenvalues determine stability regions of the 2-cluster state and conditions for its destruc-
tion. Moreover, if |ν3,N1+1| > 1 the first sub-cluster is splited and if |νN1+2,N | > 1 the second
sub-cluster is splited before the destruction.
3.1. Fixed points out of the diagonal. The fixed points can be found from the equations:
x = (1− λ)x+ (N1 − 1)λµx(1− x) +N2λµy(1− y),
y = (1− λ)y +N1λµx(1− x) + (N2 − 1)λµy(1− y).
By solving the system we get a cubic equation that can be easily solved numericaly. In the
parameters range under consideration this equation has three real roots which provide three
fixed points of the map (4), one is on the diagonal D and two others outside of it.
Fig. 4 shows regions of stability of fixed points of clusters if the ratio N1 : N2 is 1 : 9,
1 : 49, 1 : 99. Solid lines outline regions obtained by using an analytical method, and the
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
262 O. STEPANENKO, YU. MAISTRENKO, AND SH. YOUSEFI
Fig. 5. A comparison of sizes of the regions of the stabi-
lity fixed points in clusters if the ratio N1 : N2
is 1 : 9, 1 : 49, 1 : 99.
points inside the regions are calculated in a numerical simulation. The density of the points
indirectly shows the dependence of the size of the basin of attraction of fixed points on the
parameter values. The higher the density, the larger region and the higher the probability for
the trajectory with randomly chosen initial conditions to be attracted to the fixed point. Fig. 5
shows a dependence of sizes of the regions of the stability of fixed points in clusters if the ratio
N1 : N2 is 1 : 9, 1 : 49, 1 : 99 on the dimensions of the system (N = 10, 50, 100).
1. Tonu Puu Attractors, bifurcations, and chaos. Nonlinear phenomena in economics. — Berlin etc.: Springer,
2000. — 507 p.
2. Rand D. Exotic phenomena in games and duopoly models // J. Math. Econ. — 1978. — 5 . — P. 173 – 178.
3. Ahmed E., Agiza H. N. Dynamics of a Cournot game with n competitors // Chaos, Solutions and Fractals. —
1998. — 9 . — P. 1513 – 1517.
4. Kopel M. Simple and complex adjustment dynamics in Cournot duopoly models // Ibid. — 1996. —7. —
P. 2031 – 2048.
5. Agiza H. N., Bischi G. I., and Kopel M. Multistability in a dynamic Cournot game with three oligopolists //
Math. and Comput. Simulat. — 1999. — 51. — P. 63 – 90.
Received 02.07.2003
ISSN 1562-3076. Нелiнiйнi коливання, 2004, т . 7, N◦ 2
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