We consider a discrete-delay time, Kaldor non-linear business cycle model in income and capital. Given an investment function, resembling the one discussed by Rodano, we use the linear approximation analysis to state the local stability property and local bifurcations, in the parameter space. Finally, we will give some numerical examples to justify the theoretical results.business cycle; Neimark-Sacker bifurcation; discrete-delay time

Dobrescu, Loretti Isabella

Opris, Dumitru

English

Research Papers in Economics

Neimark ￿Sacker bifurcation for the discrete ￿
delay Kaldor model
Loretti I. Dobrescu￿, Dumitru Oprisy
￿ University of Padua, Department of Economics, 33 via del Santo, Padua, 35123
Italy , y West University of Timisoara, Department of Applied Mathematics, Bd.
V. Parvan, nr. 4, Timisoara, 300223 Romania
E-mail: lorettiisabella.dobrescu@unipd.it, opris@math.uvt.ro
Version: October 22, 2007
We consider a discrete ￿delay time, Kaldor non ￿linear business cycle model in income
and capital. Given an investment function, resembling the one discussed by Rodano,
we use the linear approximation analysis to state the local stability property and local
bifurcations, in the parameter space. Finally, we will give some numerical examples to
justify the theoretical results.
Key Words: business cycle, Neimark-Sacker bifurcation, discrete-delay time.
1. INTRODUCTION
Synchronization is a fundamental nonlinear phenomenon. The classical case
of synchronization consists in an external periodic (usually harmonic)signal acting
upon an auto-oscillating system with a stable cycle [3,6].
The ￿rst version of the analyzed model was proposed by Kaldor who discussed
the model of business cycle in its most simple nonlinear form. However, we cannot
take it into consideration when analyzing the case of actual dynamic economies.
The simplistic approach of Kaldor was furthermore developed by Chang and
Smith who saw the model within the dynamic systems framework through introduc-
ing the notion of continuous time expressed by a two nonlinear di⁄erential equations
system in income and capital. From this point on, the next step was to elaborate
the case of discrete-time dynamic system expressed by a system of two nonlinear
di⁄erential equations as showed by Dana and Malgrange in 1984, Hermann in 1985,
Lorenz in 1992 and 1993.
The basic concepts of Kaldor model is that if the propensity to invest is greater
than the propensity to save, then the system is unstable in a way which generates
an onset of ￿ uctuations due to the fact that if the system is far from the equilibrium
point, the propensity to invest decreases until it becomes lower than propensity to
save. Therefore, we have to take into consideration both speed of reaction to the
excess demand and the propensity to save, knowing that the ￿rst parameter has a
destabilizing e⁄ect and the second one has a stabilizing role.
Bischi [1] proposed the model to have a discrete dynamic system form, by
assuming that the ￿rms￿investment decisions are based on a expected ￿normal￿
value of income, which is exogenously given. They analyze the joint dynamic e⁄ects
of the two parameters above mentioned and show that ￿the exogenously given equi-
librium is only stable for low values of the ￿rms￿speed of reaction and su¢ ciently
1high values of the propensity to save￿ . Moreover, if the speed of adjustment is high
enough, the dynamic scenario strongly depended on the values of the propensity to
save; a low level of the propensity to save generates the situation of bi-stability.
The paper presents Kaldor model, which describes the income and capital on
n + 1 moment, taking into consideration income and capital in n, and also income
on n ￿ m, with m ￿ 0. For m = 0, the model represents the discrete-time Kaldor
model, analyzed by Bischi [2]. In section 1 we will describe the discrete - delay
Kaldor model, with respect to investment function, as presented by Rodano, and
a saving function, as considered by Keynes. We establish Jacobian matrix in a
￿x point of the model, the characteristic equation and the eigenvectors q 2 Rm+2,
p 2 Rm+2associated to the eigenvalues. In section 2 we analyze the roots of charac-
teristic equation for m = 1, function of the adjustment parameter. Using a variable
transformation, we establish that there is a value which is Neimark ￿Sacker bifur-
cation. In section 3 we describe the normal form for m = 1, as well as the orbits
of state variables. Using the software Maple11, for p;q;r ￿xed, we verify the theo-
retical results. Finally, we present the conclusions concerning the obtained results￿
utility as well as the future possible analysis of this model.
2. THE DISCRETE-DELAY KALDOR MODEL
The discrete - delay Kaldor model describes the business cycle for the state
variables characterized by income (national income) Yn and capital stock Kn, where
n 2 N. This model is represented by an equations system with discrete time and
delay, given by:
Yn+1 = Yn + s[I(Yn;Kn) ￿ S(Yn;Kn)]
Kn+1 = Kn + I(Yn￿m;Kn) ￿ qKn (1)
Where: Yn￿m represents income on moment n ￿ m, with m ￿ 1;
I : R+ ￿ R+ ! R is the investment function;
S : R+ ￿ R+ ! R is the savings function, both considered being di⁄eren-
tiable functions.
The parameter s from the set of equations (1) is an adjustment parameter
which measures the reaction of the system to the di⁄erence between investment
and saving. We admit Keynes￿ s hypothesis which states that the saving function is
proportional with income, meaning that
S(Y;K) = pY
where p 2 (0;1) is the propensity to save, with respect to income.
The investment function I is de￿ned by taking into consideration a certain
normal level of income u, and a normal level of capital stock -
pu
q , where u > 0,
q 2 (0;1).
We admit Rodano￿ s hypothesis and consider the form of the investment function
as follows:
I(Y;K) = pu + r
￿
pu
q
￿ K
￿
+ f(Y ￿ u) (2)
2where: r > 0, and
f : R ! R is a di⁄erentiable function with f(0) = 0, f0(0) 6= 0 and
f000(0) 6= 0.
The dynamic system (1), with above mentioned hypothesis, can be written as:
Yn+1 = (1 ￿ sp)Yn ￿ rsKn + sf(Yn ￿ u) + spu
￿
1 +
r
q
￿
Kn+1 = (1 ￿ s ￿ q)Kn + f(Yn￿m ￿ u) + pu
￿
1 +
r
q
￿
(3)
For m = 0 and f(x) = arctan(x) , the model was proposed and analyzed by
Bischi.
Using a change of variable
x1 = Yn￿m:::xm = Yn￿1;xm+1 = Yn;xm+2 = Kn
the application associated to system (3) is as follows:
0
B
B B
B
@
x1
:::
xm
xm+1
xm+2
1
C
C C
C
A
!
0
B
B
B B
B
B
@
x2
:::
xm+1
(1 ￿ sp)xm+1 ￿ rsxm+2 + sf(xm+1 ￿ u) + spu
￿
1 + r
q
￿
(1 ￿ r ￿ q)xm+2 + f(x1 ￿ u) + pu
￿
1 + r
q
￿
1
C
C
C C
C C
A
(4)
The ￿xed points of application (4) are the points of coordinates (y0;:::y0;k0) 2
Rm+2 where (y0;k0) is a solution of the equations system :
py + rk ￿ f(y ￿ u) ￿ pu
￿
1 +
r
q
￿
= 0
(r + q)k ￿ f(y ￿ u) ￿ pu
￿
1 +
r
q
￿
= 0 (5)
Let us consider (y0;k0) being a solution of system (5) and we note:
￿1 = f0(y0 ￿ u);￿2 = f00(y0 ￿ u);￿3 = f000(y0 ￿ u)
a10 = s(￿1 ￿ p);a01 = ￿rs;b10 = ￿1;b01 = ￿q ￿ r (6)
The following statements take place:
Proposition 1. (i). The Jacobian matrix of application (4) in the ￿x point
(y0;:::y0;k0)T is as follows:
A =
0
B
B
B B
@
0 1 ::: 0 0
::: ::: ::: ::: :::
0 ::: ::: 1 0
0 ::: ::: 1 + a10 a01
b10 ::: ::: 0 1 + b01
1
C
C
C C
A
(7)
3(ii). The characteristic equation of matrix A, given by (7), is:
￿
m+2 ￿ a(s)￿
m+1 + b(s)￿
m ￿ c(s) = 0 (8)
where a(s) = 2 + a10 + b01; b(s) = (1 + a10)(1 + b01); c(s) = a01b10
(iii). The eigenvector q 2 Rm+2, that corresponds to the eigenvalue ￿ of matrix
A has the components:
qi = ￿i￿1;i = 2;::;m + 1;qm+2 =
b10
￿ ￿ 1 ￿ b01
(9)
The eigenvector p 2 Rm+2, that corresponds to the eigenvalue ￿ of matrix AT
has the components:
p1 =
(￿ ￿ 1 ￿ a10)(￿ ￿ 1 ￿ b10)
m(￿ ￿ 1 ￿ a10)(￿ ￿ 1 ￿ b01) + ￿(2￿ ￿ 2 ￿ a10 ￿ b01)
(10)
pi = 1
￿i￿1p1; i = 2;::;m ￿ 2; pm+1 = 1
￿m￿1(￿￿1￿a10)p1; pm+2 =
￿
b10p1
The vectors q;p given by (8) and (9) satisfy the relationship
m+2 X
i=1
qipi = 1
3. THE ANALYSIS OF THE CHARACTERISTIC EQUATION IN THE FIX
POINT
We will analyze the roots of the characteristic equation (8), function of the
adjustment parameter s, for m = 1. For the m ￿ 2 analysis is quite di¢ cult to
perform.
Proposition 2. If m = 1, the following a¢ rmations are true:
(i). The equation (8) becomes :
￿
3 ￿ a(s)￿
2 + b(s)￿ ￿ c(s) = 0 (11)
(ii). The necessary and su¢ cient condition for equation (18) to admit two
complex roots with their absolute value equal to 1 and one root with absolute value
less than 1, is to exist s0 2 R so that:
jc0j < 1;ja0 ￿ c0j < 2;b0 = 1 + a0c0 ￿ c2
0 (12)
where:
a0 = a(s0);b0 = b(s0);c0 = c(s0) (13)
(iii). Let us consider:
￿1 = ￿1 ￿ p;￿1 = 2 ￿ q ￿ r;￿2 = (￿1 ￿ p)(1 ￿ q ￿ r);￿2 = 1 ￿ q ￿ r;￿3 = ￿r￿1
If, for j￿j small enough, it is satis￿ed the expression:
￿
(a0￿3 + c0￿1)(1 + ￿)
2 ￿ ￿2 (1 + ￿)
4 ￿ 2c0￿3
￿2
￿ 4
￿
￿3￿1 (1 + ￿)
2 ￿ ￿2
3
￿
￿
￿
￿
a0c0 (1 + ￿)
2 ￿ c2
0 ￿ b0 (1 + ￿)
4 + (1 + ￿)
6
￿
￿ 0 (14)
4then there is a function g : R ! R with g(0) = 0, g0(0) = 0 so that the variable
change:
s = s0 + g(￿) (15)
transforms equation (11) in equation:
￿
3 ￿ a(￿)￿
2 + b(￿)￿ ￿ c(￿) = 0 (16)
where
a(￿) = a0 + ￿1g(￿);b(￿) = b0 + ￿2g(￿);c(￿) = c0 + ￿3g(￿) (17)
Function g(￿) is given by equation:
￿
￿3￿1 (1 + ￿)
2 ￿ ￿2
3
￿
g2 +
￿
a0￿3 (1 + ￿)
2 + c0￿1 (1 + ￿)
2 ￿ 2c0￿3 ￿ ￿2 (1 + ￿)
4
￿
g +
+ a0c0 (1 + ￿)
2 ￿ c2
0 ￿ b0 (1 + ￿)
4 + (1 + ￿)
6 = 0 (18)
Equation (16), has the roots :
￿12(￿) = (1 + ￿)e￿i￿(￿);￿1(￿) =
c(￿)
(1 + ￿)
2 (19)
with:
￿(￿) = arccos
a(￿)(1 + ￿)
2 ￿ c(￿)
2(1 + ￿)
3 (20)
Following from Proposition 2, we can conclude that the analysis of the charac-
teristic equation￿roots it is made through the analysis of the equation transformed
in function of ￿. From (19) results that ￿ = 0 is the point of Neimark ￿Sacker bi-
furcation, and with (15), results that s = s0 is point of Neimark ￿Sacker bifurcation
for the given system.
4. THE NORMAL FORM OF SYSTEM (3) IF
M = 1
If m = 1, Kaldor model for which it was made the translation Y ! y + y0 and
K ! k + k0 is:
Yn+1 = ￿spyn ￿ rskn + sf(yn + y0 ￿ u) ￿ f(y0 ￿ u)
Kn+1 = ￿(r + q)kn + f(yn￿1 + y0 ￿ u) ￿ f(y0 ￿ u) (21)
Proposition 3. The following a¢ rmations are true:
(i). The normal form associated to system (21) is as follows:
zn+1 = ￿(￿)zn +
1
2
(s(￿)p2 + p3)￿2
￿
z2
n + 2znzn + zn
2￿
+
1
2
g21(￿)z2
nzn (22)
5where:
g21(￿) = ￿2 (s(￿)p2 + p3)(w20q2 + 2q2w11) + ￿3 (s(￿)p2 + p3)
w1
11 = v1
11 ￿
1
1 ￿ ￿(￿)
p2￿2 ￿
1
1 ￿ ￿(￿)
p2￿2
w2
11 = v1
11 ￿
1
1 ￿ ￿(￿)
p2q2￿2 ￿
1
1 ￿ ￿(￿)
p2q2￿2
w3
11 = v1
11 ￿
1
1 ￿ ￿(￿)
p2q3￿2 ￿
1
1 ￿ ￿(￿)
p2q3￿2
w1
02 = w1
20;w2
02 = w2
20;w3
02 = w3
20
v1
20 =
￿2
￿(￿)4 ￿ (1 + ￿1)￿(￿)2 ￿ ￿1
;v1
11 = ￿
￿2
2￿1
w1
20 = v1
20 ￿
1
￿(￿)2 ￿ ￿(￿)
p2￿2 ￿
1
￿(￿)2 ￿ ￿(￿)
p2￿2
w2
20 = ￿(￿)2v1
20 ￿
1
￿(￿)2 ￿ ￿(￿)
p2q2￿2 ￿
1
￿(￿)2 ￿ ￿(￿)
p2q2￿2
w3
20 = ￿(￿)4v1
20 ￿
1
￿(￿)2 ￿ ￿(￿)
p2q3￿2 ￿
1
￿(￿)2 ￿ ￿(￿)
p2q3￿2
q2 = ￿(￿);q3 =
￿1
￿(￿) ￿ 1 ￿ ￿1
p1 =
(￿(￿) ￿ 1 ￿ s(￿)(￿1 ￿ p))(￿(￿) ￿ 1 ￿ ￿1)
[(￿(￿) ￿ 1 ￿ s(￿)(￿1 ￿ p))(￿(￿) ￿ 1 + q + r)+
+￿(￿)
￿
2￿(￿) ￿ 2 ￿ s(￿)(￿1 ￿ p) + q + r
￿
]
p2 =
1
￿(￿) ￿ 1 ￿ s(￿)(￿1 ￿ p)
p1
p3 =
￿(￿)
￿1
p1 (23)
s(￿) is given by (15) and ￿(￿) is given by (19).
(ii). The solution of the system (21) in the neighborhood of the ￿xed point
(y0;y0;k0) 2 R3 is :
Yn = y0 + q2zn + q2zn +
1
2
w2
20z2
n + w2
11znzn +
1
2
w2
02zn
2
Kn = k0 + q3zn + q3zn +
1
2
w3
20z2
n + w3
11znzn +
1
2
w3
02zn
2
Un = y0 + q1zn + q1zn +
1
2
w1
20z2
n + w1
11znzn +
1
2
w1
02zn
2 (24)
where zn 2 C2 is a solution of equation (22).
(iii). The coe¢ cient c1(￿) associated to the equation (22) is:
c1(￿) =
 
(s(￿)p2 + p3)
2 (￿(￿) ￿ 3 + 2￿(￿))
2(￿(￿)2 ￿ ￿(￿))(￿(￿) ￿ 1)
+
js(￿)p2 + p3j
2
1 ￿ ￿(￿)
+
js(￿)p2 + p3j
2(￿(￿)2 ￿ ￿(￿))
!
￿2
2+
g21(￿)
2
(25)
Let us consider
I0 = Re(c1(0)e￿i￿(0))
6where ￿(0) is given by (20). If I0 < 0, in the neighborhood of the ￿xed point (y0;k0)
there is a a stable limit cycle (in a stable invariant closed curve).
Using the formulas from Proposition 3, using the Maple11, for ￿xed values of
p;q;r we obtain the following maps:
75. CONCLUSIONS
In this paper it is described the discrete ￿ delay Kaldor model, taking into
consideration the fact that the variation of capital depends on the value of income
8on n ￿ m moment, where n;m 2 N with m ￿ 0. For m = 1, the model obtained
is a dynamic system with discrete ￿time and delayed ￿argument. By taking as
parameter s, the adjustment parameter, we determined the value s0 for which the
characteristic equation associated to the model in the equilibrium point has complex
roots with absolute value equal to 1 and a root with absolute value less than 1, if
m = 1. Using the method of normal forms, as showed by Kuznetsov [5], we obtained
the equation which de￿nes the stable limit cycle associated to the model. Through
Maple11 we can visualize the orbits of the model￿ s variables. Therefore, in this
work, we establish for certain values of the parameters, the existence of business
cycle. For m ￿ 2, the analysis is more laborious and it will be considered in future
works. The present analysis permits us to establish the behavior of state variables
on di⁄erent moments.
REFERENCES
[1] Bischi G. I., Dieci R., Rodano G., Saltari E., ￿ Multiple attractors and global
bifurcations in a Kaldor ￿type business cycle model￿ , Journal of Evolutionary
Economics, Springer ￿Verlag 2001
[2] Bischi G. I., Valori V., ￿ Nonlinear e⁄ects in a discrete-time model of a stock
market￿ , Chaos, Solitons & Fractals, vol. 11 (2000) pp. 2103-2121
[3] Dykman G. I., Landa P. S., Neymark Y. I., ￿ Synchronizing the chaotic oscilla-
tions by external force￿ , Chaos, Solitons & Fractals, Volume 1, Issue 4, 1991,
Pages 339-353
[4] Ford N. J., Wulf V., ￿ Numerical Hopf bifurcation for the delay logistic equation￿ ,
M. C. C. M. Technical Report no. 323, Manchester University 1998
[5] Kuznetsov Y. A., ￿ Elements of applied bifurcation theory￿ , Applied Mathemat-
ical Sciences 112, Springer ￿Verlag, 1995
[6] Neimark Y. I., Landa P. S., ￿ Stochastic and chaotic osscilations￿ , Dordrecht
Kluwer Academic Publishers, 1992
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Neimark-Sacker bifurcation for the discrete-delay Kaldor model

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