Kaldor-Kalecki

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Nonlinear Analysis: Modelling and Control, 2011, Vol. 1 6, No. 2 , 191205 191

The KaldorKalecki stochastic model of business cycle

Gabriela Mircea a, Mihaela Neamtu a, Dumitru Opris b

a Faculty of Economics and Business Adminstration, b Faculty of MathematicsWest University of Timisoara

Blvd. V. Parvan 4, Timisoara 300223, Timis, [email protected]; [email protected]; [email protected]

Received:28 January 2011 / Revised: 12 May 2011 /Published online: 30 May 2011

Abstract. This paper is concerned with the deterministic and the stochastic delayed KaldorKalecki

nonlinear business cycle models of the income. They will take into consideration the investment

demand in the form suggested by Rodano. The existence of the Hopf bifurcation is studied and

the direction and the local stability of the Hopf bifurcation is also taken into consideration. For

the stochastic model, the dynamics of the mean values and the square mean values of the models

variables are set. Numerical examples are given to illustrate our theoretical results.

Keywords: nonlinear business cycle model, KaldorKalecki model, normal form, stochastic

system.

1 Introduction

The model proposed by Kaldor [1] is one of earliest and simplest nonlinear models of

business cycles. This model cannot be considered as a satisfying description of actual

economies. Nevertheless, it continues to generate a considerable amount of economic,

pedagogical and methodological interest, for the researchers in applied dynamics and

economics.

Kalecki introduced the idea that there is a time delay for investment before a business

decision. Krawiek and Sydlowski [2, 3] used the Kaleckis idea into Kaldors model and

considered the KaldorKalecki model of business cycles.

The parameters of the real models are subject to perturbations that can be considered

as stochastic or uncertain. Starting with these considerations, the associated stochastic

model can be taken into consideration.

In the present paper, we investigate the effects of the random perturbation for the

KaldorKalecki model analyzing the steady state of the model with stochastic perturba-

tion.

The analysis of this model is related to the equilibrium point. The obtained results are

connected to the stability or the existence of the limit cycle and the existence of the limit

cycle for the expected values and variances in the stochastic case.

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192 G. Mircea et al.

The reminder of the paper develops as follows. In Section 2, we describe a deter-

ministic KaldorKalecki model using the investment demand proposed in [4]. We set the

conditions for the existence of the delay parameter value for which the model displays

a Hopf bifurcation. Also, the normal form is given. In Section 3, the stochastic system

is presented and the locally asymptotic stability is analyzed according to the mean of

variables and the square mean. Numerical simulations are carried out in Section 4.

Finally, concluding remarks are given in Section 5.

2 The deterministic model of a KaldorKalecki business cycle with

delay

In the last decade, the study of delayed differential equations in business cycles has re-

ceived much attention. The first model of business cycles can be traced back to Kaldor [1],

who used a system of ordinary differential equations to study business cycles in 1940

by proposing nonlinear investment and saving functions so that the system may have

a cyclic behavior or limit cycles, which are important from the point of view of economics.

Kalecki introduced the idea that there is a time delay for investment before a business

decision. Krawiec and Szydlowski incorporated Kaleckis idea into Kaldors model by

proposing the following KaldorKalecki model of business cycles [2, 3, 5]:

Y(t) =

I

Y(t), K(t) SY(t), K(t),

K(t) =IY(t ), K(t) qK(t),(1)

whereYis the gross product,Kis the capital stock,is the adjustment coefficient in thegoods market, q(0, 1)is the depreciation rate of capital stock, I(Y, K)and S(Y, K)areinvestment and saving functions, and 0is a time lag representing delay for investmentdue to the past investment decision.

The discrete systems associated to the Kaldor and KaldorKalecki models were ana-

lyzed in [6, 7].

Consider that past investment decisions also influence the change in the capital stock

and the model (1) is extended by imposing delays in both the gross product and capital

stock [8, 9]. Thus, by adding the same delay to the capital stockK in the investmentfunctionI(Y, K)in the second equation of the system (1), the following KaldorKaleckimodel business cycles is obtained [810]:

Y(t) =

I

Y(t), K(t) SY(t), K(t),

K(t) =I

Y(t ), K(t ) qK(t), (2)As usual in a Keynesian framework, savings are assumed to be proportional to the

current level of income:

S(Y, K) =pY , (3)

where the coefficientp,p(0, 1)represents the propensity to save.

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The KaldorKalecki stochastic model of business cycle 193

As usual, the investment demand is assumed to be an increasing and sigmoid-shaped

function of the income. Without loss of generality, in what follows we shall consider the

form proposed in [4]:

I(Y, K) =pu + r

pu

q K

+ f(Y u), (4)

where puq

is the normal level of the capital stocku.In (4), two short-run investment components are considered: the first one is pro-

portional to the difference between normal capital stock and current stock, according to

a coefficientr(r >0), usually explained by the presence of adjustment costs; the secondone is an increasing, but non-linear function of the difference between current income and

its normal level. Functionfhas the propertiesf(0) = 0andf(0)= 0.This second component of the short-run investment function is a convenient specifi-

cation of the sigmoid-shaped relationship between investment and income proposed by

Kaldor. We note that this analytic specification does not compromise the generality of the

results.

From (2) with (3) and (4) we obtain the following system:

Y(t) =

pu + r

pu

q K(t)

+ f

Y(t) upY(t),

K(t) =pu + r

pu

q K(t )

+ f

Y(t ) u qK(t). (5)

The system (5) with the initial conditions:

Y() =h1(), K() =h2(), [, 0]and h1, h2 : [, 0] IR of C1-class functions, represent a system of differentialequations with delay [11].

By carrying out the translationu1(t) = Y(t) u,u2(t) = K(t) puq from (5) weget the system:

u1(t) =pu1(t) ru2(t) + f

u1(t)

,

u2(t) =qu2(t) ru2(t ) + f

u1(t )

,

u1() =h1() u, u2() =h2() puq

, [, 0], (0, 1).(6)

The linearized system of (6) in (0, 0)T

is given by:

y1(t) =a11y1(t) + a12y2(t),

y2(t) =b21y1(t ) + b22y2(t ) + a22y2(t), (7)

where

a11=(p 1), a12= r, a22=q,b21= 1, b22=r, 1= f(0).

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The characteristic function for (7) is given by:

f(, ) =2 + b + c + (d + g)e,

whereb= (p 1) + q,c = q(p 1),d= r,g = r(p 21).The analysis of the characteristic equationf(, ) = 0with respect to parameter is

done using methods from [8, 12]. This analysis leads to the following proposition:

Proposition 1. (i)If= 0and

1< min

p +

q+ r

,

p(q+ r)

q+ 2r

then equationf(, 0) = 0has roots with a negative real part.(ii) If > 0, then there exists0 > 0 so that equation f(, ) = 0 admits the roots

(0) =i0, where0 is given by:

0=

d2 + 2c b2 +

2 ,

where = (2(p 1)2 q2)2 + r4 + 6r22(p 1)2 and

0= arctan0(d

20 cd + bg)

20(g bd) + cg .

(iii) The solution of equation f(, ) = 0, denoted by = (), depends on . For

=0, = i0, we have:

M: = Re

d

d

=0, =i0

= Re

i0(di0+ g)

2i0+ b + (d di00 g0)e00

,

N: = Im

d

d

=0, =i0

.

(8)

From Proposition 1, ifM= 0 then=0 is a Hopf bifurcation.In what follows we analyze the direction and the local stability of the Hopf bifurcation

as in [13, 14]. For notational convenience, let=0+ ,(, ). Then = 0is theHopf bifurcation value of the system (6). In the study of the Hopf bifurcation problem,first we transform the system (6) into an operator equation of the form:

ut=A()ut+ R(ut), (9)whereu = (u1, u2)

T,ut= u(t + ),[, 0]. The operators A and R are defined as:

A()() =

d()d

, [, 0),A(0) + B(), = 0, (10)

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whereC1([, 0],C2),A,B are given by:

A=

a11 a12

0 a22

, B=

0 0b21 b22

(11)

and

R() =

(0, 0)T, [, 0),F1(, ), F2(, )

T, = 0,

(12)

F1(, ) =12

221(0)

1

63

31(0),

F2(, ) =1

2 221()

1

6 331().

(13)

ForC1([0, ],C2), the adjoint operator A ofA is defined as:

A()(s) =d(s)

ds , s[0, ),

AT(0) + BT(), s= ,

ForC1([, 0),C2)andC1([0, ],C2)define the bilinear form:

, = T(0)(0) 0

1

0

T( )B() d

d. (14)

To determine the Poincare normal form of the operator A(), we need to calculate theeigenvector ofA associated with the eigenvalue1= i0 and the eigenvector ofAassociated with the eigenvalue2= 1.

Using (10), (11) we obtain:

Proposition 2. (i)The eigenvectorofA associated with the eigenvalue1 is given by:() =ve1, [, 0],

wherev = (v1, v2)T,v1= a12,v2= 1a11.(ii)The eigenvector ofA associated with the eigenvalue2= 1 is given by:

(s) =we2s, s[0, ],

where

w= (w1, w2)T, w1= h, w2= ,

h= b21e

1

2a11 , = 1

(h b21)v1+ (1 hb22)v2.

(iii)With respect to(14)we have:(s), ()

= 1,

(s),()

=

(s), ()

= 0,

(s),()

= 1.



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Next, we construct the coordinates of the center of the manifold 0 at = 0, [13].Let

z(t) =, ut, w(t, ) =ut2 Re

z(t)()

.

On the center manifold0, we considerw(t, ) =w(z(t),z(t), ), where

w(z,z, ) =w20()z2

2 + w11()zz+ w02()

z2

2 +

andz,z are the local coordinates of the center manifold 0 in the direction of and ,respectively. Notice that for = 0, for the solutionut0 of (9), we have:

z(t) =1z(t) +

,Rw(t, ) + 2Re z(t)().We rewrite this as z(t) = 1z(t) + g(z, z)on the center manifold 0 in the powers

ofz and z:

g(z,z) =g20z2

2 + g11zz+ g02

z2

2. (15)

From (12), (13) and (15) we obtain:

Proposition 3. For the system(6) we have:

g20= w1F120+ w2F220, g11= w1F111+ w2F211,(16)

g02= w1F102+ w2F202, g21= w1F121+ w2F221,

where

F120=2v21 , F102= F120, F111=2v21 ,F220= 2v

21, F202=

F220, F211= 2v21 ,

F121=2

w120(0)v1+ 2w111(0)v1 3v21 v1,

F221=2

w220(0)v1+ 2w211(0)v1 3v21v1,

and

w20() =

w120(), w220()T

, w11() =

w111(), w211()T

are given by:

w20() =g20

1ve1 g20

31ve2 + E1e

21,

w11() =g11

1ve1 g11

1ve2 + E2, [, 0),

E1=

A + e220B 21I1

F20, E2=(A + B)F11,F20= (F120, F220)

T, F11= (F111, F211)T.

(17)

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Let

D1= det

A + e220B21I

, D2= det(A + B)

and

d111= a22 21+ b22e220

D1, d112=

a12D1

, d121=b21e

220

D1, d122=

a1121D1

,

d211=a22+ b22

D2, d212=

a12D2

, d221=b21D2

, d222= a11D2

.

From (17) we have:

E11= (d111

d112)2a

212, E12= (d

121

d122)2a

212,

E21= (d211d212)2a212, E22= (d221d222)2a212,

g20= (1 h)a212, g11= (1 h)a212, g02= (1 h)a212= g20,(18)

w120(0) = g20

1a12 g02

31a12+ E11, w111(0) =

g111

a12 g111

a12+ E21,

w220(0) = g201

(1a11)e20 g2031

(2a11)e10 + E12e220 ,

w211(0) = g111

(1a11)e20 g111

(2 a11)e10 + E22.

(19)

From (16) and (19) we obtain:

g21= hF121+ F221, (20)

where

F121=2a12

g201

a12 g0231

a12+ 2g111

a12 2 g111

a12+ E11+ 2E21

3a312,

F221=2a12

g201

(1 a11)e20 g2031

(2 a11)e10

+ 2g111

(1 a11)e20 2 g111

(2 a11)e10

+ 2E22+ E12e220

3a312.

(21)

Therefore, we can compute the following parameters:

C(0) = i

20

g20g11 2|g11|2 1

3|g02|2

+

g212

,

2=Re(C(0))M

, 2= 2 Re(C(0)), T0=Im(C(0)) + 2N0

,

(22)

whereM andNare given by (8).



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In the formulas (22), 2 determines the direction of the Hopf bifurcation; 2 deter-mines the stability of the bifurcation periodic solutions; T0 determines the period of thebifurcating periodic solution.

Proposition 4. (See [12, 13].) (i) If2 > 0 (< 0) the Hopf bifurcation is supercritical(subcritical)and the bifurcating periodic solution exists for > 0(< 0).

(ii)If0< 0 (>0)the solutions are orbitally stable(unstable).(iii)IfT0> 0 ( 0,2> 0.The solution of (23) is a stochastic process denoted by Y(t) = Y(t, ), K(t) =K(t, ), . From the Chebyshev inequality the possible rang ofY, Kat a timetis almost determined by its mean and variance at time t. So, the first and the secondmoments are important for investigating the solutions behavior.

Consider the stochastic system given by (23). Linearizing (23) around the equilibrium

(u, pu/q)yields the linear stochastic differential delay equation:

dy(t) =

A2y(t) + B2y(t )

dt Cy(t) dw(t), (24)

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wherey(t) = (y1(t), y2(t))T and

A2=

h11 h12

0 h22

, B2=

0 0k21 k22

, C=

1 00 2

.

h11= (1p), h12=r, h22=q, k21= 1, k22=r.Lety(t)be the fundamental solution of the system:

y(t) =A2y(t) + B2y(t ).

The solution of (23) is a stochastic precess given by:

y(t, ) =y(t)t

0

Y(t s)Cy (t , ) dw(s),

wherey(t)is the solution given by:

y(t) =Y(t)(0) +

0

Y(t s)(s) ds

and : [, 0]IR2 is the family of continuous functions.The existence and uniqueness theorem for the stochastic differential delay equation

has been established in [15].The solutiony(t, ) is a stochastic process with the distribution at any time t, deter-

mined by the initial function(). From the Chebyshev inequality, the possible rangeofy , at the time t is almost determined by its mean and variance at the time t. Thus,the first and second moments of the solutions are important for the investigation of the

solutions behavior.

We have used Eto denote the mathematical expectation and we denote y (t, ) byy(t). From (24), we obtain:

Proposition 5. (i)The moments of the solution for the system(24)are given by:

E

y(t)

=A2E

y(t)

+ B2E

y(t )

, (25)

whereE(y(t)) = (E(y1(t)), E(y2(t)))T.(ii)The characteristic equation of the system(25)is given by:

2 +

(p 1) + q

+ q(p 1) +

r (p 1) + r1

e = 0. (26)

(iii)If = 0, the roots of the equation (26)have a negative real part if and only if1satisfies the relation:

p(1 q)2 q < 1 0,b22d22> 0, wherea2= (p 1) + q, b2= q(p 1), c2= r, d2= r1(p 1),

then there exists0 and0 so that0 =i0, =0 is a solution of the equation(26);0 is a positive solution of the following equation:

4 +

a222b2 c2

2 + b22 d22= 0and

0= 1

0arctan

0a2d2+ 0c2(20 b2)d2(22

b2)

20c2a2

. (27)

From (27) we have = ()andRe (0) = 0and d()d |=0,=i0= 0.

Thus, 0is a Hopf bifurcation. The solutions of the system (25) on the center manifoldare given by:

E(t) =z(t)(0) + z(t)(0),

where

(0) = (h12, 0h11)T, 0= i0,and

z(t) =0z(t), z(t) =x(t) + iy(t).

For [0, 0)the mean values of the variables for the system (24) are asymptoticallystable; for > 0, they are unstable and while for =0they are periodical.

To examine the stability of the second moments of y(t) for the linear stochasticdifferential delay equation (24) we use the Ito rule. We have:

d

dtE

y(t)yT(t)

=E

dy(t)yT(t) + y(t) dyT(t) + Cy (t)yT(t)C

=E

A2y(t)yT(t) + y(t)yT(t)A2+ B2y(t )yT(t)

+ y(t)yT(t )BT2 + Cy(t)yT(t)C

. (28)

Let R(t, s) = E(y(t)yT(s)) be the covariance matrix of the process y(t) so thatR(t, t)satisfies:

R(t, t) =A2R(t, t) + R(t, t)AT2 + B2R(t , t)

+ R(t, t

)BT2

+ CR(t, t)C. (29)

From (29),R(t, s) = (Rij(t, s))i,j=1,2and Rij(t, s) =E(yi(t)yj(s))we obtain:

Proposition 6. (i)The differential system(29)is given by:

R11(t, t) =

2h11+ 21

R11(t, t) + 2h12R12(t, t ),

R22(t, t) =22R22(t, t) + k22R22(t, t ) + k21R12(t, t ),

R12(t, t) = (h12+ h22+ 12)R12(t, t) + k21R11(t, t ) + k22R12(t, t ).(30)

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(ii)The characteristic function of (30)is given by:

f2(, ) =

2 22+ re

42 221+ 12+ 2(1p) r+ 2(1p)(12r r) + 2r1e

. (31)

For the proof of Proposition 6(ii), considerRij(t, s) = e(t+s)Kij , i = 1, 2, where

Kij are constants. ReplacingRij(t, s)in (30) and setting the condition that the resultingsystem we should accept nontrivial solution, we obtain f2(, ) = 0.

The analysis of the second moments is done studying the roots of the characteristic

equationf2(, ) = 0.From (31) we have:

Proposition 7. (i) If = 0, the roots of the characteristic equationf2(, ) = 0 havenegative real parts if and only if

22 < r, H 1< 1< H2, (32)

where

H1=2

224r8(1p)

2 , H2=

2+

22+4r8(1p)2

.

(ii)If= 0, 22 < rthen for(0, 1)the roots of the equation222+re = 0have negative real parts, where

1= 11

arctan2122

, 1=

r2

42

2 .

(iii)If= 0and the relations(32)hold, then for(0, 2)the roots of the equation:22 + a1 + b1+ c1e

= 0,

where

a1=

21+ 12+ 2(1p) r

,

b1= (1p)(12 r r), c1= r1have negative real parts, where

2= 12

arctan a12222 b1

, 2=

4b

2

1 a2

1+

(a

2

1 4b2

1)2

+ 16c

2

14

.

Because the solution of the equationf2(, ) = 0is = ()from (31) we have:

Re (i) = Re

d()

d

=i, =ii

= 0, i= 1, 2.

Thus, =i,i = 1, 2is a Hopf bifurcation.



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If(0, 1)the square mean values are asymptotically stable. For =1 we obtain theorbits(t, R11(t, t)),(t, R22(t, t)),(t, R12(t, t))given in Fig. 4, Fig. 5 and Fig. 6:

Fig. 4.(t, R11(t, t)). Fig. 5.(t, R22(t, t)). Fig. 6.(t, R12(t, t)).

For0= 4.58, in Fig. 7 and Fig. 8 are displayed the orbits of the mean values.

Fig. 7.(t, E11(t, t)). Fig. 8.(t, E22(t, t)).

For (0, 3.18) the square mean values and the mean values are asymptoticallystable.

The numerical simulations verify the theoretical results. Also, we can consider the

functionsf(x) = 0.02sin x,f(x) = 0.2 arctan(x).

5 Conclusions

The analysis of a KaldorKalecki business cycle model in this paper allowed us to ob-

tain some new dynamic scenarios which may be interesting for researchers in applied

dynamics and economics.

The paper has analyzed the KaldorKalecki model and the equilibrium point of the

model with stochastic perturbation.



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We have determined the values of the delay for which the KaldorKalecki system is

asymptotically stable and for which the system displays a limit cycle. The direction and

stability of the bifurcating periodic solutions are determined.

For the stochastic model, we have analyzed the square mean and the variance of the

models variables.

We have determined the values offor which the square mean values and the vari-ances are stable.

As in [16] the hybrid KaldorKalecki model will be taken into consideration in our

next paper.

Acknowledgements

The research was done under the Grant with the title The qualitative analysis and nu-

merical simulation for some economic models which contain evasion and corruption,

CNCSIS-UEFISCU Romania (Grant No. 1085/2008).

The authors would like to express their gratitude for valuable comments on this manu-

script from the anonymous referee.

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