Bifurcations in economic growth model with distributed time delay transformed to ODE
BBifurcations in economic growth model withdistributed time delay transformed to ODE
Luca Guerrini
Department of Management, Polytechnic University of Marche,Piazza Martelli 8, I–60121, Ancona (AN), Italy;e-mail: [email protected]
Adam Krawiec
Institute of Economics, Finance and Management, Jagiellonian University,(cid:32)Lojasiewicza 4, 30-348 Krak´ow, Poland;e-mail: [email protected]
Marek Szyd(cid:32)lowski
Astronomical Observatory, Jagiellonian University,Orla 171, 30-244 Krak´ow, Poland;Mark Kac Complex Systems Research Centre, Jagiellonian University,(cid:32)Lojasiewicza 11, 30-348 Krak´ow, Poland;email: [email protected]
Abstract
We consider the model of economic growth with time delayed in-vestment function. Assuming the investment is time distributed wecan use the linear chain trick technique to transform delay differentialequation system to equivalent system of ordinary differential system(ODE). The time delay parameter is a mean time delay of gammadistribution. We reduce the system with distribution delay to boththree and four-dimensional ODEs. We study the Hopf bifurcation inthese systems with respect to two parameters: the time delay param-eter and the rate of growth parameter. We derive the results from a r X i v : . [ ec on . T H ] F e b he analytical as well as numerical investigations. From the former weobtain the sufficient criteria on the existence and stability of a limitcycle solution through the Hopf bifurcation. In numerical studies withthe Dana and Malgrange investment function we found two Hopf bi-furcations with respect to the rate growth parameter and detect theexistence of stable long-period cycles in the economy. We find that de-pending on the time delay and adjustment speed parameters the rangeof admissible values of the rate of growth parameter breaks down intothree intervals. First we have stable focus, then the limit cycle andagain the stable solution with two Hopf bifurcations. Such behaviourappears for some middle interval of admissible range of values of therate of growth parameter. Keywords : Kaldor-Kalecki growth model Distributed time delay Bi-furcation analysis Hopf bifurcation Linear chain trick
In economics many processes depends on past events, so it is natural to usethe time delay differential equations to model economic phenomena. Twomain areas of applications are business cycle and economic growth theories.In last decades the study of impact of investment delay was a subject of andetailed scrutiny as a mechanism for endogenous cycle to describe businesscycles and growth cycle. One of the most influential models of businesscycle with time delay was the Kaldor-Kalecki model [1] which was based onthe Kaldor model, one earliest endogenous business cycle model [2, 3]. TheKaldor is a prototype of a dynamical system with cyclic behaviour in whichnonlinearity plays a crucial role to generate the endogenous cycles. Thenonlinearities are common feature used to model the complexity of economicsystems [4]. In turn, the investment delay was assumed to be the averagetime of making investment as it was proposed by Kalecki [5]. Apart from thepioneering interest in time delay in economics, the time delay systems werestudied in many domains of science [6].The Kaldor-Kalecki business cycle model was a subject of many studiesas well as augmentations. This model was modified by incorporating theexponential trend to describe growth of an economy [7]. This new Kaldor-Kalecki growth model was formulated as similar way as the Kaldor growthmodel was obtained from the Kaldor business cycle model [8].The models with distributed delays are a more realistic description of2conomic systems with time delay. There are some examples of such modelsin context of economic growth [9].In this paper we modify the Kaldor-Kalecki growth model to allow fordistributed time of investment. Instead of average time of investment comple-tion the distributed time length of investment is considered. This approachprovides more realistic characteristic of investment processes. This growthmodel has the form of the dynamical system with a distributed time delay.The unimodal distribution function for investment is assumed.While the delay differential equations methods are developing rapidly, themathematical methods for ordinary differential equations are superlative, es-pecially when distributed delays are considered. Therefore, it is convenientto approximate a system with distributed delay with a system of ordinarydifferential equations. One of the method to reduce a system with the dis-tributed delay to an ordinary differential equation system is the linear chaintrick [10, 11, 12]. In consequence infinite-dimensional dynamical systems areapproximated by finite-dimensional dynamical system, where the dimensionof the system can be chosen.The main aim of the paper is to study possible bifurcation due to changeof the parameter values of the Kaldor-Kalecki growth model. We considertwo simplest cases of three and four dimensional dynamical systems obtainedthrough the linear chain trick from the Kaldor-Kalecki growth model withdistributed delay. For both models we establish the conditions for existenceof the Hopf bifurcation with respect to the time delay parameter and the rateof growth parameter. We show the both parameters play role in a scenarioleading to the Hopf bifurcation and arising cyclic behaviour.In the numerical part of the paper we determine in details the ranges ofparameter values for which cyclical behaviour is possible. For these analysiswe use the investment function obtained by Dana and Malgrange for Frencheconomy [8]. As in theoretical part of the paper we choose the time delayparameter and the rate of growth parameter. Additionally we consider theadjustment parameter. We show that the combination of these three param-eters of model can lead to arising the cycles through Hopf. In the space ofthese three parameters of the model we obtain the surface (a section of aparaboloid) separating the regions with stable and cyclic solutions.3
Model In Economic growth cycles driven by investment delay , Krawiec and Szyd(cid:32)lowski[7] formulated the model based on the Kaldor business cycle model with twomodifications: exponential growth introduced by Dana and Malgrange [8]and Kaleckian investment time delay [1]. This model of economic growth isdescribed by the following system of differential equations with time delay τ ≥
0, ˙ y ( t ) = α [ I ( y ( t ) , k ( t )) − γy ( t ) + G ] − gy ( t ) , (1)˙ k ( t ) = I ( y ( t − τ ) , k ( t )) − ( g + δ ) k ( t ) , (2)where I ( y ( t ) , k ( t )) = k ( t )Φ( y ( t ) /k ( t )) , withΦ( y/k ) = c + d1 + e − a ( vy/k − , and α, γ, g, δ, G , g and a, c, d , v are positive constants. It can be found thatthe system has a unique fixed point ( y ∗ , k ∗ ) with positive coordinates, where y ∗ = x ∗ k ∗ and k ∗ = αG gx ∗ + α [ sx ∗ − ( g + δ )] , with x ∗ the unique solution of the equationΦ( x ∗ ) = g + δ. Because of the S-shape of function Φ( x ) , we have that x ∗ always exists and thevalues of y ∗ and k ∗ depend only on x ∗ (in our case, c < g + δ < c + d). Noticethat, for economic considerations, the investment function I ( y ( t ) , k ( t )) issuch that I ∗ y = I y ( y ∗ , k ∗ ) = a d v e − a ( vx ∗ − [1 + e − a ( vx ∗ − ] > I ∗ k = I k ( y ∗ , k ∗ ) = g + δ − x ∗ I ∗ y < . (4)In this paper, we generalize their model by replacing the time delay in(2) with a distributed delay as follows,˙ y ( t ) = α [ I ( y ( t ) , k ( t )) − γy ( t ) + G ] − gy ( t ) , (5)˙ k ( t ) = I t (cid:90) −∞ y ( r ) κ ( t − r )d r, k ( t ) − ( g + δ ) k ( t ) , (6)4here κ ( · ) is a gamma distribution, i.e. κ ( ξ ) = (cid:16) mT (cid:17) m ξ m − e − mT ξ ( m − , with m a positive integer that determines the shape of the weighting func-tion. T ≥ T → m = 1 (weak delay kernel) and m = 2 (strong delay kernel). Using the so-called linear chain trick technique[12], system (5)-(6) can be transformed into equivalent systems of ODEs.More precisely, defining the new variable u ( t ) = t (cid:90) −∞ y ( r ) (cid:18) T (cid:19) e − T ( t − r ) d r one has the system (case m = 1)˙ y ( t ) = α [ I ( y ( t ) , k ( t )) − γy ( t ) + G ] − gy ( t ) , (7)˙ u ( t ) = 1 T [ y ( t ) − u ( t )] , (8)˙ k ( t ) = I ( u ( t ) , k ( t )) − ( g + δ ) k ( t ) , (9)while defining the new variables p ( t ) = t (cid:90) −∞ y ( r ) (cid:18) T (cid:19) ( t − r )e − T ( t − r ) d r and w ( t ) = t (cid:90) −∞ y ( r ) (cid:18) T (cid:19) e − T ( t − r ) d r, one obtains the system (case m = 2)˙ y ( t ) = α [ I ( y ( t ) , k ( t )) − γy ( t ) + G ] − gy ( t ) , (10)˙ p ( t ) = 2 T [ w ( t ) − p ( t )] , (11)˙ w ( t ) = 2 T [ y ( t ) − w ( t ]) , (12)˙ k ( t ) = I ( p ( t ) , k ( t )) − ( g + δ ) k ( t ) . (13)5e will now analyse the stability and Hopf bifurcation of systems (7)-(9)and (10)-(13) by determining eigenvalues of linearised systems around thecritical point ( y ∗ , y ∗ , k ∗ ) and ( y ∗ , y ∗ , y ∗ , k ∗ ), respectively. m = 1 The characteristic equation of the linearised system (7)-(9) at the criticalpoint ( y ∗ , u ∗ , k ∗ ) , where u ∗ = y ∗ , is given by (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) αI ∗ y − αγ − g − λ αI ∗ k T − T − λ I ∗ y I ∗ k − ( g + δ ) − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , (14)where λ denotes a characteristic root. A direct calculation implies that (14)leads to λ + a ( T ) λ + a ( T ) λ + a ( T ) = 0 , (15)where a ( T ) = 1 T − A, a ( T ) = − AT − B, a ( T ) = 1 T (cid:0) − B − αI ∗ k I ∗ y (cid:1) . with A = α (cid:0) I ∗ y − γ (cid:1) − g − x ∗ I ∗ y and B = (cid:2) α (cid:0) I ∗ y − γ (cid:1) − g (cid:3) x ∗ I ∗ y . The necessary and sufficient condition for the local stability of the equilibriumpoint is that all characteristic roots of (15) have negative real parts, which,from the Routh–Hurwitz condition, is equivalent to a ( T ) > , a ( T ) > a ( T ) a ( T ) > a ( T ) . Then, a ( T ) > A < . In fact, by contradiction, if A = 0, then a ( T ) = − (cid:2) x ∗ I ∗ y (cid:3) < . Onthe other hand, if
A >
0, then
B > , and so a ( T ) < . The fact
A < a ( T ) > a ( T ) > B + αI ∗ k I ∗ y < . Thus, a ( T ) > ≤ , and it is verified for g + δ − ( g + αγ ) x ∗ < B > . Finally, letus consider a ( T ) a ( T ) > a ( T ) . Since a ( T ) a ( T ) − a ( T ) = ( AB ) T + ( A + αI ∗ k I ∗ y ) T − AT , the sign of a ( T ) a ( T ) − a ( T ) depends on the sign of ( AB ) T + ( A + αI ∗ k I ∗ y ) T − A, which is a quadratic polynomial in T. We have now severalcases.i) If B = 0 , then a ( T ) a ( T ) − a ( T ) > A + αI ∗ k I ∗ y ≥ A + αI ∗ k I ∗ y < T < A/ ( A + αI ∗ k I ∗ y ). I ) = T ∗ .ii) If B > , then AB < − A > . By Descartes’ rule of signs wefind that the polynomial ( AB ) T + ( A + αI ∗ k I ∗ y ) T − A has exactly onepositive root T = T ∗ . Hence, a ( T ) a ( T ) − a ( T ) > < T < T ∗ . iii) If B < , then AB > − A > . Applying again the Descartes’ ruleof signs we see that ( AB ) T + ( A + αI ∗ k I ∗ y ) T − A has (two) sign changesonly if A + αI ∗ k I ∗ y <
0, meaning that this polynomial may have twopositive roots T ∗ < T ∗ . If this happens, then a ( T ) a ( T ) − a ( T ) > < T < T ∗ and T > T ∗ .Let T = T ∗ such that a ( T ∗ ) a ( T ∗ ) − a ( T ∗ ) = 0 , namely T ∗ = T ∗ j ( j =0 , , , . The curve T = T ∗ divides the parameter space into stable andunstable parts. Choosing T as a bifurcation parameter, we apply the Hopfbifurcation theorem to establish the existence of a cyclical movement. Thistheorem asserts the existence of the closed orbit, if the characteristic equation(15) has a pair of pure imaginary roots and a non-zero real root, and if thereal part of the imaginary roots is not stationary with respect to the changesof the parameter T . At the critical value T = T ∗ , Eq. (15) factors as[ λ + a ( T ∗ )] (cid:2) λ + a ( T ∗ ) (cid:3) = 0 , so we have the following three roots λ , = ± i (cid:112) a ( T ∗ ) = ± iω ∗ and λ = − a ( T ∗ ) < . Next, let us investigate the sign of the real parts of this rootsas T varies. A differentiation of (15) with respect to T yields (cid:2) λ + 2 a ( T ) λ + a ( T ) (cid:3) d λ d T = − (cid:2) a (cid:48) ( T ) λ + a (cid:48) ( T ) λ + a (cid:48) ( T ) (cid:3) , (16)7here a (cid:48) ( T ) = − T < ,a (cid:48) ( T ) = AT < ,a (cid:48) ( T ) = − T (cid:0) − B − αI ∗ k I ∗ y (cid:1) = − a ( T ) T < . Then, from (16), we get R e (cid:18) d λ d T (cid:19) T = T ∗ = − a (cid:48) ( T ∗ ) a ( T ∗ ) − a ( T ∗ ) a (cid:48) ( T ∗ ) + a (cid:48) ( T ∗ )2 [ a ( T ∗ ) + a ( T ∗ )] . Since − a (cid:48) ( T ∗ ) a ( T ∗ ) − a ( T ∗ ) a (cid:48) ( T ∗ ) + a (cid:48) ( T ∗ ) = − AT ∗ (cid:0) BT ∗ + 1 (cid:1) , we obtain sign (cid:20) R e (cid:18) d λ d T (cid:19) T = T ∗ (cid:21) = sign (cid:0) BT ∗ + 1 (cid:1) . If B ≥ , we observe that R e (d λ/ d T ) T = T ∗ > T ∗ = T ∗ , T ∗ ) holdsalways true, whether if B < , then R e (d λ/ d T ) T = T ∗ > T ∗ = T ∗ , T ∗ )if 0 < T < / √− B, and R e (d λ/ d T ) T = T ∗ < T > / √− B. The previous analysis leads to the following conclusions.
Theorem 1
Let
A < , with A defined as in (15) . If B = 0 and A + αI ∗ k I ∗ y < or if B > and B + αI ∗ k I ∗ y < , thenthere exists T = T ∗ > such that the equilibrium point ( y ∗ , y ∗ , k ∗ ) of (7) - (9) is locally asymptotically stable for all T < T ∗ and unstable for T > T ∗ . System (7) - (9) undergoes a Hopf bifurcation at ( y ∗ , y ∗ , k ∗ ) when T = T ∗ . If B < , then there exists < T ∗ < T ∗ such that the equilibrium point ( y ∗ , y ∗ , k ∗ ) of (7) - (9) is locally asymptotically stable for all T < T ∗ and T < T ∗ , and unstable for all T ∗ < T < T ∗ . A comparison of / √− B with T ∗ and T ∗ yields that system (7) - (9) undergoes a Hopf bifurcationat ( y ∗ , y ∗ , k ∗ ) when T = T ∗ or T = T ∗ or T = T ∗ and T = T ∗ . For system (7)-(9) figure 1 presents the bifurcation diagram for the timedelay parameter T . 8 Ty HB Figure 1: The bifurcation diagram for model for system (7)-(9) m = 1 withinvestment function (25) for delay parameter T . The solid line indicates acritical point with asymptotic stability and the dot-dash line corresponds toan unstable critical point with a limit cycle around it. Case m = 2 The characteristic equation of the linearised system (10)-(13) at the criticalpoint ( y ∗ , p ∗ , w ∗ , k ∗ ) , where p ∗ = w ∗ = y ∗ , takes the form (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) αI ∗ y − αγ − g − λ αI ∗ k − T − λ T T − T − λ I ∗ y I ∗ k − ( g + δ ) − λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (17)9here I ∗ y and I ∗ k are defined as in (3) and (4), which leads to the followingfourth order algebraic equation in λλ + a ( T ) λ + a ( T ) λ + a ( T ) λ + a ( T ) = 0 , (18)where a ( T ) = 4 T − ( M + N ) , a ( T ) = 4 T − M + N ) T + M N, and a ( T ) = 4 T (cid:20) M N − M + NT (cid:21) , a ( T ) = 4( M N + P ) T , with M = α (cid:0) I ∗ y − γ (cid:1) − g, N = I ∗ k − ( g + δ ) < , P = − αI ∗ k I ∗ y > . (19)According to the Routh–Hurwitz conditions for stable roots, the equilibriumpoint ( y ∗ , y ∗ , y ∗ , k ∗ ) of system (14) is locally asymptotically stable if a ( T ) > , a ( T ) > , a ( T ) > a ( T ) a ( T ) a ( T ) > a ( T ) + a ( T ) a ( T ) , namelyif 4 T − ( M + N ) > , M N − M + NT > , M N + P > ϕ ( T ) = (cid:2) ( M + N )( M N ) (cid:3) T + [( M + N ) ( P − M N )] T + (cid:8) M + N ) (cid:2) ( M + N ) + 2 M N − P (cid:3)(cid:9) T + (cid:8) (cid:2) P − ( M + N ) (cid:3)(cid:9) T + 16( M + N ) < . Taking in mind that
N <
P > , we derive that conditions (3) holdalways true if M ≤ . On the other hand, when
M > , they are valid if M + N < , M N + P >
T < ( M + N )( / ( M N ) . The condition ϕ ( T ) < M = 0 . In fact, in this case, ϕ ( T ) = (cid:0) N P (cid:1) T + (cid:2) N (cid:0) N − P (cid:1)(cid:3) T + (cid:2) (cid:0) P − N (cid:1)(cid:3) T + 16 N is such that ϕ (0) < ϕ (+ ∞ ) = + ∞ . Hence, there exists at least a posi-tive value of T, say T ∗ , such that ϕ ( T ) = 0 . We need now to recall Descartes’10ule of signs and its corollary, that state “the number of positive roots of thepolynomial ϕ ( T ) is either equal to the number of sign differences betweenconsecutive nonzero coefficients, or is less than it by an even number” and“the number of negative roots is the number of sign changes after multiply-ing the coefficients of odd-power terms by −
1, or fewer than it by an evennumber”, respectively. Applying these rules to the polynomial ϕ ( T ) , we getthat ϕ ( T ) has one positive zero and the number of negative zeros must beeither 2 or 0 . Therefore, ϕ ( T ) < T < T ∗ . When M = 0 , the equilibriumpoint ( y ∗ , y ∗ , y ∗ , k ∗ ) of (14) is locally asymptotically stable for T < T ∗ .Assume there exists T ∗ > ϕ ( T ∗ ) = 0 , i.e. a ( T ∗ ) a ( T ∗ ) a ( T ∗ ) − a ( T ∗ ) − a ( T ∗ ) a ( T ∗ ) = 0 . In this case, we can rewrite the characteristic equation (18) as (cid:2) a ( T ∗ ) λ + a ( T ∗ ) (cid:3) (cid:2) a ( T ∗ ) λ + a ( T ∗ ) λ + a ( T ∗ ) a ( T ∗ ) − a ( T ∗ ) (cid:3) = 0 . so that we have two purely imaginary roots λ , = ± i (cid:115) a ( T ∗ ) a ( T ∗ ) = ± iω ∗ , and two other roots, λ , = − a ( T ∗ ) ± (cid:112) a ( T ∗ ) − a ( T ∗ ) [ a ( T ∗ ) a ( T ∗ ) − a ( T ∗ )]2 a ( T ∗ ) , which have real parts different from zero since λ + λ = − a ( T ∗ ) < λ λ = [ a ( T ∗ ) a ( T ∗ ) − a ( T ∗ )] /a ( T ∗ ) > . Differentiating the characteristic equation (18) with respect to T , we have (cid:2) λ + 3 a ( T ) λ + 2 a ( T ) λ + a ( T ) (cid:3) d λ d T = − (cid:2) a (cid:48) ( T ) λ + a (cid:48) ( T ) λ + a (cid:48) ( T ) λ + a (cid:48) ( T ) (cid:3) , (20)i.e. d λ d T = − a (cid:48) ( T ) λ + a (cid:48) ( T ) λ + a (cid:48) ( T ) λ + a (cid:48) ( T )4 λ + 3 a ( T ) λ + 2 a ( T ) λ + a ( T ) , (21)11here a (cid:48) ( T ) = − T , a (cid:48) ( T ) = − T + 4( M + N ) T , and a (cid:48) ( T ) = − M NT + 8( M + N ) T , a (cid:48) ( T ) = − M N + P ) T . Letting λ = iω ∗ in (21), a direct calculation yields R e (cid:18) d λ d T (cid:19) T = T ∗ = − a ( T ∗ ) ϕ (cid:48) ( T ∗ )2 (cid:8) a ( T ∗ ) a ( T ∗ ) + [ a ( T ∗ ) a ( T ∗ ) − a ( T ∗ )] (cid:9) , where ϕ (cid:48) ( T ∗ ) = a (cid:48) ( T ∗ ) a ( T ∗ ) a ( T ∗ ) + a ( T ∗ ) a (cid:48) ( T ∗ ) a ( T ∗ ) + a ( T ∗ ) a ( T ∗ ) a (cid:48) ( T ∗ ) − a ( T ∗ ) a (cid:48) ( T ∗ ) − a ( T ∗ ) a (cid:48) ( T ∗ ) a ( T ∗ ) − a ( T ∗ ) a (cid:48) ( T ∗ ) . Let us notice that sign (cid:2)
Re (d λ/ d T ) T = T ∗ (cid:3) = sign [ − ϕ (cid:48) ( T ∗ )], and recall thatsign (cid:2) Re (d λ/ d T ) T = T ∗ (cid:3) > (cid:2) Re (d λ/ d T ) T = T ∗ (cid:3) < Theorem 2
Let M be defined as in (19) . Let M = 0 . There exists T ∗ > such that the equilibrium point ( y ∗ , y ∗ , y ∗ , k ∗ ) of (14) is locally asymptotically stable for T < T ∗ , unsta-ble for T > T ∗ , and bifurcates to a limit cycle through a Hopf bifurcationat the equilibrium point when T = T ∗ . Let M (cid:54) = 0 . The equilibrium point ( y ∗ , y ∗ , y ∗ , k ∗ ) of (14 ) is locallyasymptotically stable if M < and ϕ ( T ) < or if M > , M + N < ,M N + P > , T < ( M + N )( / ( M N ) and ϕ ( T ) < . If there exists T = T ∗ such that ϕ ( T ∗ ) = 0 and ϕ (cid:48) ( T ∗ ) (cid:54) = 0 , then a Hopf bifurcationmay occurs at the equilibrium point as T passes through T ∗ . The rate of growth bifurcation analysis
Let us consider the dynamics of the system (7)-(9) with respect to the changeof the parameter g (the rate of economic growth). Proposition 1
The critical point of system (7)-(9) (and equivalently system(1)-(2) always exists for the rate of growth parameter g in the interval c − δ For real eigenvalues we have the following proposition Proposition 3 In the interval c − δ < g < c +d − δ , there are two subintervalswith the positive values of the discriminant (24) there two cases for the valuesof rate of growth parameter g . In these subintervals there are three negativereal eigenvalues. The subintervals of the parameter g with two negative and one positiveeigenvalues are non-physical regions as the critical point ( y ∗ , u ∗ , k ∗ ) does notlie in a positive quadrant.For complex eigenvalues we have the following proposition Proposition 4 In the interval c − δ < g < c + d − δ and negative values ofthe discriminant (24) for the increasing value of the rate of growth parameter g there are two supercritical Hopf bifurcations. For the value g = g , Hopf thelimit cycle is created, and for the value g = g , Hopf the limit cycle is destroyed( g , Hopf < g , Hopf ). Therefore, as the the rate of growth parameter is increasing in the interval g min = c − δ < g < c + d − δ = g max the eigenvalues change as follows. In thefirst subinterval ( g min ; g ) there are three real eigenvalues (two negative, onepositive). In the second subinterval ( g ; g , Hopf ) there are three real eigenval-ues (three negative). In the third subinterval ( g , Hopf ; g , Hopf ) there are onereal eigenvalue (negative) and one conjugate complex eigenvalue (positivereal parts). In the fourth subinterval ( g , Hopf ; g ) there are three real eigen-values (three negative). And finally, in the fifth subinterval g , ( g max ) thereare three real eigenvalues (two negative, one positive).We some example values of parameters we can determine the values ofthe rate of growth parameter for which the eigenvalues change their characteror sign. We assume the values of investment function parameters obtainedby Dana and Malgrange, namely, c = 0 . 01, d = 0 . a = 9, v = 4 . 23. Wefix also the following model parameters α = 1, γ = 0 . δ = 0 . G = 2and T = 1. The rate of growth parameter g is taken within the interval g min = c − δ < g < c + d − δ = g max . The results are presented in table 1.For system (7)-(9) figure 2 presents the bifurcation diagram for the rateof growth parameter g . 14able 1: The intervals of values of rate of growth parameter g and respectivesigns of eigenvalues of the characteristic equation (22). It is assumed that c = 0 . 01, d = 0 . a = 9, v = 4 . 23 (the investment function), α = 1, γ = 0 . δ = 0 . G = 2 and T = 1 (rest model parameters).real eigenvalues complex eigenvalues rate of growth parameter1 negative pair with negative real part (0.003, 0.0101198)1 negative pair with positive real part (0.0101199, 0.0203258)1 negative pair with negative real part (0.0203259, 0.029) The original Kaldor model exhibited the limit cycle behaviour due to theHopf bifurcation caused by the increase of the parameter α value [3]. Later,it has been augmented both by introducing the investment lag T and exoge-nous growth trend g . The increase of the investment time delay parametervalue also generates the limit cycle [1]. However, the dependence the Hopfbifurcation on the rate of growth parameter was not elaborated so far. BothChang and Smyth in the Kaldor model [3] as well as Dana and Malgrange inthe Kaldor model with exogenous growth trend investigated the parameter α as the bifurcation parameter[8].We conduct the numerical analysis of both models: m = 1 and m = 2.We assume the investment function with the following parameter values: c = 0 . 01, d = 0 . a = 9 and v = 4 . 23 [8] I ( y, k ) = k Φ( y, k ) = 0 . 01 + 0 . − . y/k − . (25)We also assume the following the model parameters: γ = 0 . δ = 0 . G = 2 and consider the three parameters in the following intervals: T ∈ (0 , α ∈ (0 . , . 0) and g ∈ (0 . , . Case m = 1 In this case we consider the three-dimensional model (7)-(9) for the statevariables ( y, u, k ). In this model we study numerically the stability of thecritical point ( y ∗ = u ∗ , k ∗ ) to find the values of parameter T for which the15 HB HB y g Figure 2: The bifurcation diagram for model for system (7)-(9) m = 1 withinvestment function (25) for delay parameter T . The solid line indicatescritical point with asymptotic stability and the dot-dash line corresponds tothe unstable critical with a limit cycle around it.critical point loses the stability and the limit cycle is created through the Hopfbifurcation mechanism. We study in details the dependence of the bifurcationvalue of T on the model parameters α and g as well as the dependence thebifurcation value of g on the model parameter α and T .The bifurcation surface in the parameter space ( a, g, T ) is presented infigure 3. The region below the surface corresponds to the asymptotic stabilityof the critical point ( y ∗ , u ∗ , k ∗ ). The region inside corresponds to parametervalues for which system (4) has an unstable critical point with a limit cyclearound it.Let conduct more detailed analysis and consider relations between twoparameters with a third parameter fixed. First, the relation of T on α for g = 0 . 016 is shown in Fig. 4. We find that the asymptotic stability regionexist only if α < . g = 0 . α ∈ (0 . , . Figure 3: The Hopf bifurcation surface in the space of parameters ( α, g, T )for system m = 1 and with investment function (25). Outside of the surfaceis the region of asymptotic stability, while inside of the surface is the regionof parameters values for which a limit cycle solution exists.we find the relation T bi ( α ) is T bi = − . . α . (26)Second, we analyze the dependence of the parameter T on the parameter g with the fixed value of α . We consider the three values of parameter α .The stability regions on the plane ( g, T ) are shown for α = 0 . α = 0 . g increases the region B is growing.The bifurcation line separating the region A and B is described by aquadratic equation T bi = a g + a g + a . (27)Now, consider the time paths of the model for the economic variable y for different values of the time delay parameter for the amplitude and periodof cycles. In region I there is a stable equilibrium reached by trajectories in17 .60 0.65 0.70 0.75 0.80 . . . . . . . α T region I (stable) region II (unstable) Figure 4: The plane of parameters ( α, T ) for system m = 1 and g = 0 . α = 0 . , . y ( t )obtained for given T = 0 . , . , y ( t ) = 1, k ( t ) = 100 where t ∈ ( − T, T increases. Case m = 2 In this case we consider the four-dimensional model (10)-(13) for the statevariables ( y, p, w, k ). In this model the critical point values of ( y ∗ = p ∗ = w ∗ , k ∗ ) are the same as the critical point values of ( y ∗ = u ∗ , k ∗ ) of threedimensional model presented in the previous section.In a similar way as in the previous section we analyse the occurrence ofthe Hopf bifurcation for the parameter T depending on parameters α and g .The relation of T on α for g = 0 . 016 is shown in Fig. 7. The asymptoticstability region exist only if α < . g = 0 . m = 1.18 .010 0.012 0.014 0.016 0.018 0.020 g T region Aregion B 0.010 0.012 0.014 0.016 0.018 0.020 g T region A region B region A Figure 5: The plane of parameters ( g, T ) for system m = 1 with investmentfunction (25). Here, it is assumed that α = 0 . α = 0 . T bi ( g ).Let us study the cycle characteristics for some values of the parameter α with different values of the delay parameter T . Figure 8 shows the solu-tions of y for two cases of α = 0 . , . T = 0 . , . , . 0. The amplitude and period of cycles are decreasing as theparameter T increases for α = 0 . Comparing m = 1 and m = 2 The models considered can be treated as the approximation of the delayKaldor-Kalecki growth model. It is important to find how good the subse-quent approximations are. Therefore, we compare the bifurcation values ofthe parameter T in models m = 1 and m = 2.First, we consider the bifurcation diagram in the parameter plane ( α, T )presented in Fig. 9. We find that for the given value of parameter α the Hopfbifurcation value of the parameter T bi is lower for the model m = 2. Thisdifference is zero for T = 0 and then increases as the value T bi increases for19 200 400 600 t y T = 0.5T = 1.5T = 3.0 0 200 400 600 t y T = 0.5T = 1.5T = 3 Figure 6: Trajectories of model m = 1 with investment function (25) forthe parameter g = 0 . 016 and α = 0 . α = 0 . α . On the other hand, for the fixed valueof the parameter T , the bifurcation value of the parameter α bi is greater inthe model m = 2. For the parameter g = 0 . 016 the difference is close zero at α = 0 . α = 0 . g = 0 . . 234 at α = 0 . 6. It is demonstrated in Fig. 9 for g = 0 . g = 0 . 011 (right panel).We compare trajectories of y ( t ) for systems m = 1 and m = 2 with thesame initial conditions. In Figs. 10, 11 there are trajectories y ( t ) for assumedthe parameter g = 0 . 016 and combinations of parameters α and T . We findthat for the same parameters α , g and T the period of cycles is smaller formodel m = 2 and amplitude is also smaller, although the difference is verysmall. For example, for α = 0 . g = 0 . 016 and T = 3, the period of cycle inmodel m = 1 is 114 . 85 and in model m = 2 is 116 . 45 while amplitudes are12 . . m we shouldobtain the cycles with longer periods and amplitudes.We can explore further approximations of the model and compare valuesof the bifurcation parameter g for successive m in table 2.20 .60 0.65 0.70 0.75 0.80 . . . . . . . α T region I (stable) region II (unstable) Figure 7: The plane of parameters ( α, T ) for system m = 2 and parameter g = 0 . 016 with investment function (25). Region I is the region of asymptoticstability, while region II and III are regions of parameters value for which alimit cycle solution exists. We study the Kaldor-Kalecki growth model with distributed delay transformto the ordinary differential equation system using the linear chain trick. Itallows to choose the arbitrary dimension of the system. We consider two caseswhere the system is transformed to three-dimensional and four-dimensionaldynamical systems.The model possesses the equilibrium for some interval of values of rate ofgrowth parameter. The interval depends on the values of investment functionwhich is taken to model analysis.We study a bifurcation to a limit cycle (the Hopf bifurcation) due tothe change of two parameters: the time delay T and the rate of growth g . Additionally we take take into consideration the speed of adjustmentparameter α which is the bifurcation parameter of original Kaldor model.For the better insight of dynamics we analyzed the bifurcations in the thethree dimensional space of these parameters.When we consider the dynamics of the model under the change of thegrowth rate parameter, we discover numerically two bifurcation values of the21 200 400 600 t y T = 0.5T = 1.5T = 3.0 0 200 400 600 t y T = 0.5T = 1.5T = 3 Figure 8: Trajectories of model m = 2 with investment function (25) forparameter g = 0 . 016 and α = 0 . α = 0 . g bi,1 and g bi,2 for subsequent approxi-mations model g bi,1 = 1 g bi,2 = 2 m m m m .60 0.65 0.70 0.75 . . . . . . . α T region I (stable) region II (unstable)m=1m=2 α T region I (stable) region II (unstable)m=1m=2 Figure 9: The plane of parameters ( α, T ) for system ( m = 1) and ( m = 2)with investment function (25). Here, it is assumed that g = 0 . 016 (left panel)and g = 0 . 011 (right panel). The dashed line is for model m = 1 and thedotted line is for model m = 2. These bifurcation curves separates Region Iof asymptotic stability on the left side of curves and region II of limit cyclesolution on the right side of curves.There are two oscillating regimes. For lower and higher rates of growththe oscillations are damped and asymptotically stationary state is reached.For intermediate rates of growth the self-sustained oscillations of constantamplitude are present. For some model parameters this intermediate inter-val of rate of growth values is obtained to be (0 . , . α , γ and δ .All numerical analyses have been done with Dana and Malgrange’s in-vestment function for the French macroeconomic data [8]. • The Kaldor-Kalecki model with distributed delay is reduced to theordinary differential system using the linear chain trick technique. • Depending on the value of the parameter m of the Γ distribution func-tion the reduced system is ( m + 2)-dimensional ordinary differentialequation system. 23 200 400 600 t y m=1m=2 0 200 400 600 t y m=1m=2 Figure 10: Trajectories of models m = 1 and m = 2 with investmentfunction (25) for the same initial condition ( y = p = w = 15 , k = 100) (leftpanel) for α = 0 . g = 0 . T = 0 . T = 3. • For the increasing time delay parameter there is the supercritical Hopfbifurcation. • For the increasing rate of growth parameter, first the limit cycle emergesand then the limit cycle disappears. Therefore, there are two supercrit-ical Hopf bifurcations with two bifurcation values of the rate of growthparameter. • For some values of parameters α and T , in the allowed range of therate of growth parameter values, both for lower and higher values ofthe rate growth parameter the model has the stable stationary pointwhile for the middle range of parameter values there is the limit cycle. • The period of cycle increases and decreases as the rate of growth pa-rameter increases in the range of unstable solution. • Comparing the models with different m the stable region in the param-eter space is slightly diminished as m is greater.24 200 400 600 t y m=1m=2 0 200 400 600 t y m=1m=2 Figure 11: Trajectories of models m = 1 and m = 2 with investmentfunction (25) for the same initial condition ( y = p = w = 15 , k = 100) (leftpanel) for α = 0 . g = 0 . T = 0 . T = 3. 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