THE DISCRETE – DELAY KALDOR-KALECKI MODEL
WITH HOPF BIFURCATION
Dumitru Opris, PhD Professor at WestUniversity of Timisoara, Faculty of Mathematics
Loretti Dobrescu, Assistant lecturer at West University of Timisoara, Faculty of Economic Sciences, PhD student at University of Padova, Italy
Address: Università degli Studi di Padova, Dipartimento Marco Fanno, via del Santo, 33, 35123, Padova, Italy
Telephone+39 335 5633801
Fax: +39 049 827 4211
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JEL classifications: C, E
Abstract. We will take into consideration a non-linear, Kaldor – Kalecki type, discrete – delay time business cycle model in income and capital. Being given an investment function, which will follow the model described by Rodano, we use the linear approximation analysis to state the local stability property and local bifurcations, in the parameter space. In the final part of the paper, some numerical examples will be given in order to make possible the justification of the theoretical results.
Keyword: business cycle, Hopf bifurcation, discrete-delay time.
INTRODUCTION
The analysed model in its initial version was proposed by Kaldor, who took into consideration the most simple nonlinear form of the model of business cycle. Despite this approach, this form cannot be considered when analysing the case of dynamic economies nowadays.
The simplistic manner of analysing the model of Kaldor was furthermore developed by Chang and Smith, who overcame the static approach and translated the model the dynamic systems framework through introducing the notion of continuous time expressed by a two nonlinear differential equations system in income and capital. From this stage on, the next step was to elaborate the case of discrete-time dynamical system, expressed by a system of two nonlinear differential 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 fluctuations 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 first parameter has a destabilizing effect and the second one has a stabilizing role.
Bischi proposed the model to have a discrete dynamical system form, by assuming that the firms’ investment decisions are based on a expected “normal” value of income, which is exogenously given. They analyze the joint dynamic effects of the two parameters above mentioned and show that “the exogenously given equilibrium is only stable for low values of the firms’ speed of reaction and sufficiently high 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 moment, taking into consideration income and capital in , and also income on , with . For , the model represents the discrete-time Kaldor model, analyzed by Bischi. 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 Jacobi matrix in a fix point of the model, the characteristic equation and the eigenvectors, associated to the eigenvalues. In section 2 we analyze the roots of characteristic equation for and cases, function of the adjustment parameter. Using a variable transformation, we establish that there is a value which is Neimark – Sacker bifurcation. In section 3 we describe the normal form of the Kaldor model for , and in section 4 we describe the normal form for , as well as the orbits of state variables. Using the software of Maple 9, for fixed, we verify the theoretical results. Finally, we present the conclusions concerning the obtained results’ utility as well as the future possible analysis of this model.
- The discrete - delay Kaldor model
The discrete - delay Kaldor model describes the business cycle for the state variables characterized by income (national income) and capital stock , where . This model is represented by an equations system with discrete time and delay, given by:
(1)
Where: represents income on moment, with ;
is the investment function;
is the savings function, both considered beingdifferentiable functions.
The parameter from (1) is an adjustment parameter which measures the reaction of the system to the difference between investment and saving. We admit Keynes’s hypothesis which states that the saving function is proportional with income, meaning that, where is the propensity to save, with respect to income.
The investment function is defined by taking into consideration a certain normal level of income – , and a normal level of capital stock - , where , . We admit Rodano’s hypothesis and consider the form of the investment function as follows:
(2)
where: , and
is a differentiable function with , and .
The dynamic system (1), with above mentioned hypothesis, can be written as:
(3)
For and , the model was proposed and analyzed by Bischi.
Using a change of variable the application associated to system (3) is as follows:
(4)
The fixed points of application (4) are the points of coordinates where is a solution of the equations system :
(5)
Let us consider being a solution of system (5) and we note:
, ,
(6)
, , ,
The following statements take place:
Proposition 1. (i). The Jacobi matrix of application (4) in the fix point is as follows:
(7)
(ii). The characteristic equation of matrix , given by (7), is:
(8)
where: , ,
(iii). The eigenvector, that corresponds to the eigenvalue of matrix has the components:
(8) , ,
The eigenvector , that corresponds to the eigenvalue of matrix has the components:
(9)
, , ,
The vectors given by (8) and (9) satisfy the relationship
- 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 , for and cases. For the analysis is quite difficult to perform.
Proposition 2. If , the following affirmations are true:
(i). The characteristic equation (8) is as follows:
(10)
(ii). If , and ,
where
(11)
then equation (10) has the complex roots in absolute value equal with 1.
(iii). Taking into consideration the variable change:
(12)
where: , , .
Equation (10) can be written as:
(13)
where:
(14)
Equation (13) has the roots :
(15) ,
for small enough.
(iv). The eigenvector which corresponds to the eigenvalue , considering the matrix is:
(16) ,
The eigenvector which corresponds to the eigenvalue , considering the matrix is:
(17) ,
Proposition 3. If , the following affirmations are true:
(i). The equation (8) is :
(18)
(ii). The necessary and sufficient 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 so that:
(19) , ,
where:
(20) , , .
(iii). Let us consider:
, , , ,
If, for small enough, it is satisfied the expression:
(21)
then there is a functionwith , so that the variable change:
(22)
transforms equation (18) in equation:
(23)
where:
(24) , ,
Function is given by equation:
(25)
Equation (23), has the roots :
(26) ,
with:
(27)
Following from Proposition 2 and Proposition 3, we can conclude that the analysis of the characteristic equation’ roots it is made through the analysis of the equation transformed in function of . From (15) and (26) results that is the point of Neimark – Sacker bifurcation, and with (22), results that is point of Neimark – Sacker bifurcation for the given system.
- The normal form of system (3) if
If the Kaldor model in which we made the translation and is:
(28)
Proposition 4. Using the method from Kuznetsov and Ford-Wulf, the following affirmations are true:
(i). The canonic form of system (28) is as follows:
(29)
where is given by (15), is given by (12), is given by (17) and
(ii). The coefficient associated to form (29) is:
(30)
(iii). Let us consider , where is given by (15). If in the neighbourhood of the fixed point there is a closed stable curve. If , the curve is closed but unstable.
(iv). The solution of Kaldor system (4), in the neighbourhood of the fixed point is:
(31) ,
where , is a solution of equation (21). The investment function and the savings function are given by:
(32) ,
Using the formulas from Proposition 4, through the software called Maple 9, for fixed values of we obtain the following results:
If q:=0.2,r=0.4,p=0.5,s0 given by (11) and f(x):=tan(x),then the trajectory is represented in the phase plane (Y,K) and l1(0)=0.36315, as it can be noticed in the following map:
- The normal form of system (3) if
If , Kaldor model for which it was made the translation and is:
(33)
Proposition 5. The following affirmations are true:
(i). The normal form associated to system (33) is as follows:
(34)
where:
(35)
is given by (22) and is given by (26).
(ii). The solution of the system (33) in the neighbourhood of the fixed point is :
(36) ,
,
,
where k1.
with the investment function and the savings function :
,
where is a solution of equation (34).
(iii). The coefficient associated to the equation (34) is:
(37)
Let us consider where is given by (27). If , in the neighbourhood of the fixed point there is a stable invariant closed curve.
Using the formulas from Proposition 5, through the software called Maple 9, for fixed values of we obtain the following maps:
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 on moment, where with . For , the model is the discrete – time Kaldor model. For , the model obtained is a dynamic system with discrete – time and delayed – argument. By taking as parameter , the adjustment parameter, we determined the value for which the characteristic equation associated to the model in the equilibrium point has complex roots with absolute value equal to 1, if , and complex roots with absolute value equal to 1 and a root with absolute value less than 1, if . Using the method of normal forms, as showed by Y. A. Kuznetsov, we obtained the equation which defines the invariant curve associated to the model. Through Maple9 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, the analysis is more laborious and it will be considered in future works. The present analysis permits us to establish the bahaviour of state variables on different moments.
REFERENCES
- G. I. Bischi, R. Dieci, G. Rodano, E. Saltari, ‘Multiple attractors and global bifurcations in a Kaldor – type business cycle model’, Journal of Evolutionary Economics, Springer – Verlag 2001
- N. J. Ford, V. Wulf, ‘Numerical Hopf bifurcation for the delay logistic equation’, M. C. C. M. Technical Report no. 323, Manchester University 1998
- Y. A. Kuznetsov, ‘Elements of applied bifurcation theory’, Applied Mathematical Sciences 112, Springer – Verlag, 1995