Abstract
This paper studies continuous nonlinear economic dynamics with a continuous delay of a Kaldor type modified in dimension two. The important results are, on the one hand, the boundedness of solutions, the existence of an attractive set, and the permanence of the system and, on the other hand, the local and global stability of equilibrium points.
1. Model
The complex nonlinear dynamic has been introduced into the analysis of economic phenomena to explain not only the fluctuations observed in the series studies but also the economic crisis in the capitalist system. Thus, economists such as Goodwin (1967) and Kaldor (1955-1956) have employed dynamic models to explain that the cyclic and chaotic growth curves are the economic phenomena endogenous to the capitalist system itself. Within the framework of structural reforms for economic dynamics, NINDJIN et al. in [1] suggest a proven model originating from ecology which could have a certain range in analysis and regulation of financial systems. This model of dimension two describes how the GDP and an economic capital interact in accordance with the model of Kaldor modified in order to increase the resilience of these dynamics against possible disruptions. Mathematical analysis of this model (see [1]) demonstrates that it is bounded and permanent and admits under certain circumstances an attractive set. On the one hand, this permanence manifests itself in the form of stationary growth of capital stock and product (stable interior equilibrium point). On the other hand, it appears in the capital cyclic growth and the product (limit cyclic). So, a capital stock rupture or a long-term production is prevented. It is also shown that the financial system stability (relative to the capital and the product) is overall; i.e., it does not depend on either stock level or production at the initial time. Facing a possible disruption of one of the control parameters of the economic system, we have analyzed, in [2], the model bifurcations. We have shown that the model admits a transcritical bifurcation, a pitchfork bifurcation or a Hopf bifurcation. In the last one, when the model is disrupted, it changes from a stationary balanced growth to a cyclic growth by preventing a GDP or capital crisis. Interactions between the GDP and the economy’s capital could not be clearly and definitively understood or explained regardless of past situations which may affect the present or the future. In this paper, we are going to focus on the way in which investments are evolved and savings are established. Indeed, economies finance their investments through their own savings or those of other economies via financial structures with an interest rate. Regarding the savings, it is known that they are made up of a portion of profits or wages. Thus, when the economy is able to be self-funding by its savings, then the net profit increases (i.e., profit, deprived of the portion set aside to pay interest, debt, ). So, saving at a time depends on the net profit; consequently, the GDP from a time where Let us suppose that is the deadline needed by this saving to reach a certain threshold likely to ensure the self-financing of the investments of the economy.
Let us consider the dynamics with no delay of the modified Kaldor type and the following assumption:
with and .
denotes the product, denotes the stock of capital, and and indicate, respectively, the growth rate of the product and the stock of capital depending on the following economic parameters:(i) the trend of rate of increase in GDP for a given (future) period in absence (or in neglect) of the losses,(ii) the maximum (monetary) value of GDP we can get from this economy for that given period,(iii) the currency adjustment factor,(iv) the maximum (monetary) value of the genuine saving of this economy for the given period,(v) the maximum (monetary) value of the saving supported by the economy in the given period,(vi) the maximum value of the investment rates losses for the given period,(vii) the maximum capital stock for the given period,(viii) the derivative relative to the capital of the investment rate in the absence (or in the neglect) of the losses (when ) for the given period,(ix) the investment rate when the capital is null () and this rate has suffered no loss () for the given period,(x) the share of the GDP converted into stock of capital for the given period,(xi) the accumulation rate of capital when the product is null () and that the investment rate has suffered no loss () for the given period.
(see [1], page 4).
Assumption 1. The ratio saving-capital at a given time depends on the GDP produced since with a probability of exponential lag (see, [3], appendix, page 272).
So, the ratio saving-capital with delay is with so that . Then,
By replacing by , one obtains continuous nonlinear economic dynamics with a continuous delay of a Kaldor type modified in dimension two. Enclosing (1) to the system with a continuous delay, one gets the following system with discrete delay in dimension three:
with , , and the feedback function is consequently
To facilitate the qualitative study of system which possesses parameters , let us change the variables by reducing the number of parameters.
Let us define the new variables:
Let us define the new parameters of control: , and Then, system becomes
with ∈ and = so that
In the long run, we shall adopt the following notations:
2. Boundedness and Equilibria Points of
2.1. Boundedness of the Solutions of Model
Lemma 2. The interior and the boundary of a positive cone are invariant for model .
Proof. Given . Let be in . One knows that Thus, on the one hand, ∈ ⇒ ∈, and, on the other hand, ∈ ⇒ ∈.
Lemma 3 (see [4]). Given and a continuous and derivable function as there is verifying . Then, ,
Theorem 4. Let us assume that . Let us pose and, then, model is permanent. Otherwise, the set defined byWhich is a bounded set, positively invariant for model .
Proof. Let us consider system and Lemma 3. (1)One has the following: does not depend on the variable . So, one obtains the same results like those of the model with no delay, i.e., and (2)One has the following: because Then, with if and = ; + + if and, in that case, − = By property (6) of Lemma 3, one obtains ≤ (3)One knows that Let us suppose that and, then, there exist so that, , Let us suppose that = Thus, for any very tiny, there is so that, , ≥ Let us pose = and Then, , < ≤ . Thus Hence, , > for = is decreasing upon On the one hand, it is known that = and . Meanwhile, ≤ × and, then, ≤ = . On the other hand, = + − and + ≥ = + + + Then, , and ≥ − − So, with and = − Therefore, applying property (7) of Lemma 3 one obtains ≥ . Consequently, ≥ for So, one obtains If then − ≥ Thus one obtains ≥ = with ≤ or + ≤ (4)It is known that ≥ × So, ≥
Remark 5. Let us consider the notations of Theorem 4. (1)Let us consider the following assumptions: One remarks that each of assumptions (13) and (14) implies, respectively, condition (9).(2)If , one has So, we can adjust the minimum value of by choosing the value of
2.2. The Equilibria Points of Model
2.2.1. Points of Trivial Equilibria
If then system admits two points of trivial equilibria:
If then system admits three points of trivial equilibria: , , and
2.2.2. Points of Interior Equilibria
Given such as
(1)System does not admit any point of interior equilibria if .(2)Any interior equilibrium point of system verifies the following relations:
Designating the projection over the plan , we study the border dynamics of the model.
3. Border Dynamics on the Plan
with .
The equilibria points of model areLet us give below the results on the permanence of the model. Therefore, let us consider the notations of Theorem 4. If and then model is permanent. Among others, the set defined bywhich is a bounded set, positively invariant for model .
3.1. Local Stability of the Equilibria Points of Model
Performing the spectral study of Jacobian matrix of the system linearized around each of the points of equilibrium, one obtains, classically, the following conclusions:(1)Stability of :(a) is an unstable node if .(b) is a point unstable saddle repulsive following direction and attractive following direction if (2)Stability :(a) is stable if .(b) is a point unstable saddle attractive following direction and repulsive following direction if .(3)Stability of for :(a) is stable if .(b) is a point of unstable saddle if , repulsive following , and attractive following Consider(4)Given interior equilibrium point of testifying the system (19)-(20) and J its associated Jacobian matrix.(a) is stable (stable node or stable focus) if and .(b) is marginal or center if and .(c) is unstable if or ( and ), precisely:(i) is a node or a focus if and ,(ii) is a point unstable saddle if .
With the trace and determinant of J, vectors and .
3.2. Global Stability of
Now let us define the conditions for which the stability of the product and the stock of capital of the economy is global; i.e., it does not depend on the produced quantities and level of stock at the initial moment. For this study, we define an appropriate Lyapunov function.
Theorem 6. Let us consider the following assumptions:where and are defined in Theorem 4.
If assumptions (25), (26), (27), (28), (29), (30), (31), (35), (36), and (37) are verified, then, the unique point of the interior equilibrium of model is globally and asymptotically stable.
Proof. Let us consider system . Let us suppose that assumption (35) is verified and, then, model admits a unique interior equilibrium point verifyingLet us note that . Given and one hasConsidering the Lyapunov functional one hasLet us pose By using relation (29) and model , one obtains with and , .
Let us overestimate and and, then, and (1)One has , , if .(2)if then So, Therefore, if Consequently, = < < Thus, and if , , and Hence, , , if The model is permanent if Then, Moreover, So, from formulas (35), one obtains (27) ⇒ (36) and (28) ⇒ (37). Hence, , , if assumptions (25)–(28) are verified.
Remark 7. As , the unique interior equilibrium point of model is globally and asymptotically stable if the following assumptions are verified:with constant and defined in Theorem 4.
In fact, it is worth taking the Lyapunov functional so that with and so thatConsidering (38)–(40), one shows, similarly in , that ,
4. Border Dynamics of Plan
with The equilibria of model areLet , , , and The model is permanent and the set defined byis limited, positively invariant.
Concerning the local stability, one obtains, for any delay , the following results:(1) is an unstable saddle point, repulsive along the direction , and attractive along the direction .(2) is stable.
The global stability of equilibrium is given in the following theorem.
Theorem 8. and, then, the unique interior equilibrium point of model is globally and asymptotically stable.
Proof. Let us consider with and . Let us pose
and . Then, one obtains the following system:Let us note that .
Then, the superior derivative of in relation to time, alongside the solutions of (46), givesLet us note that . Considering , one has and . Hence, the superior derivative of in relation to time, alongside solutions (46), givesConsidering the functional: thenLet us pose Thus,Let Then,So, if Consequently, if and .
5. Border Dynamics of the Plan
with and
The equilibria of model are as follows: for ,Model is nonpermanent. In fact,
Let us now give the results upon stability of the equilibria of model .(1)The equilibrium is stable if and a point unstable saddle if (2)The equilibrium for is stable. Moreover, the equilibrium is globally and asymptotically stable. In effect, and
6. Local Stability of Model
Let us pose that , , and for withThe system linearized around the point of equilibrium , iswithThe characteristic equation of (57) is
6.1. Local Stability of and
We are in situations where Let us use the following criterion Routh-Hurwitz: the interior equilibrium is stable if , , , and and unstable if not.
Theorem 9. Let us suppose that Then, the equilibria and of model are all unstable.
Proof. Let us consider formula (63)–(67). (1)The equation of is One has the following: and Let us suppose that and Then, one would have Consequently, which is absurd. Hence, the equilibrium of model is unstable by application of the Routh-Hurwitz criterion.(2)One has . Meanwhile, and, then, so the equilibrium of model is unstable by application of Routh-Hurwitz criterion.
6.2. Local Stability of and
We are in the situation where . In order to assess the influence of delay over the local stability of model , we use the results upon the local stability of model obtained taking into model and the results of [5] upon the local stability of the delay systems. Let us consider the characteristic equation (63).(i)One has the following: Then, and do not have any common imaginary roots.(ii)One has the following: and are polynomials with real coefficients so and (iii)If so (iv)Let us pose Then, when
Let us consider the function defined on by the following:
Let us determine the sign of the real part of a solution of the characteristic equation
Lemma 10. Let us consider , a solution of the characteristic equation , with Given positive root of and the associated delay certifying the following relation:Let us pose Then,with
Proof. Given positive root of Let us assume that there is a delay so thatLet us poseOne has Therefore, and . Thus, .and, meanwhile, Then,
with , = + , = + , = + − − + , and = + + −
It is noticed that the analysis of the local stability of the system with delay when depends on the existence of positive root for function The coefficients and , of , depend on Therefore, its positive root depends on Using the Cartan method and Viet formulas, one proves on the one hand that admits at least a positive root if On the other hand, if , then, admits either of two positive roots whereas does not admit any positive root.
The condition is equivalent to the following assumption:
Let us denote , , and Then, one obtains the following proposition.
Proposition 11. Let us suppose that
Let us pose and , (1) admits at least a positive root if(2) does not admit any positive root if one of the following conditions is verified:
Now, let us use the results of Theorem 4 from [5] to study the stability of these points of equilibriums and .
Theorem 12. One posesLet us consider one of these assumptions (82)–(85) and one of the following conditions:With defined in formula (23), consider the following. (1)Let us suppose that one of assumptions (83)–(85) is verified for i=2,3. Then, there is no change of stability for equilibrium .(2)Let us suppose that assumptions (82) and one of these assumptions (87)–(89) are verified for i=2.(a)If then the equilibrium is unstable for any (b)If then the equilibrium is unstable if and stable if (3)Let us suppose that assumption (82) is verified for i=3.(a)In case ,(i)If assumption (90) is verified then the equilibrium is stable if and unstable if (ii)If assumption (90) is not verified then the equilibrium is unstable for any (b)Considering ,(i)If assumption (90) is verified then the equilibrium is stable for every (ii)If assumption (90) is not verified then the equilibrium is unstable if and stable if With and , are the denotations defined in Lemma 10.
Proof. Let us consider formula (63)–(67). Let us consider the function defined in relation to the formula (70)-(71) for any point of equilibrium , Let us consider assumptions (82). Then, exists so that, for any , admits a positive root (see Proposition 11). Considering the associated delay determined from Lemma 10, consider the following:(1)Let us suppose that one of assumptions (83)–(85) is verified in i=2,3. Then, , does not admit any positive root. Thus, there is no change of stability of the equilibrium (2)Stability of : let us consider that one of assumptions (87)-(88) is verified. Then, If assumption (89) is verified, then, Hence, if assumptions (87)-(88) are verified. Then, let us apply Theorem 4 from [5]. So, let us suppose that . It is known that and so is unstable because for Taking into account the conclusions of Theorem 4 from [5], one obtains the stability of (3)Let us suppose that the assumption of (90) is verified. Then, so Thus, let us apply Theorem 4 from [5]. It is known that when and then is stable if assumption (90) is verified and unstable if not. Taking into account the conclusions of Theorem 4 from [5], one gets the stability of
7. Global Stability of Model
Theorem 13. Let us suppose that model is permanent and that it admits a unique interior equilibrium point. Ifthen the unique interior equilibrium point of model is globally and asymptotically stable.
Proof. Let us consider model and a unique interior equilibrium point. Let us pose , , and .
Then, , , and Thus, Let us denote . Consider Then, . Meanwhile, so the superior derivative of in relation to the time, alongside the solutions of system , gives Let us denote . However, if Then,Let us note . One has and . Then, superior derivative of in relation to the time, alongside the solutions of , results inLet us consider the functional ; then,Let us pose that Thus,Let us pose Then, from formulas (95), (96), and (100), one obtainsOne has Model is permanent. So, on the one hand, one has if In addition, Moreover,
Then, if
So, if assumptions (91)–(93) are checked.
Corollary 14. For , if the following assumptions are verified:then, the unique interior equilibrium point of model is globally and asymptotically stable.
Proof. Let us consider the assumptions of Theorem 13.The model is permanent. ThenAssumptions (91) and (93) give the following assumption:Moreover, assumptions (91) and (92) give the following assumption:Posing , , , and
Then, condition (113) becomes By imposing the condition, , one obtains and assumptions (103)–(109). According to Theorem 13, the unique interior equilibrium point of model is globally and asymptotically stable.
8. Numerical Simulations
8.1. Limit Cycle
Let us consider the period and the following control parameters: , , and
For the graphic illustration, let us consider the initial conditions , the step , and the numbers iterations of time One obtains the maximum values of , , and :
The trivial equilibrium points of the model are ,
, and
In addition, = (2.19899664564055; 2.19198952607149; 0.00121680786072133) is the unique interior equilibrium. So, the illustrating Figure 1 is for the initial time .
(i) Interpretation. The trajectories of GDP and of capital and (the effect of delay) are periodic. The orbit starting from the initial condition gravitates around the equilibrium point without never reaching it. There is, therefore, a limit cycle around this equilibrium point . The orbit revolves around the equilibrium point. They are moving away from trivial equilibrium points and converge towards the limit cycle around Indeed, one has and According to Proposition 11, is the positive root of One has and So, is unstable according to Theorem 12 because for The trivial equilibrium points of the model are all unstable for Hence, we have the graphic illustration.
8.2. Global Stability
Let us consider the period and the following control parameters: = (101390.241; 50695.121; 0.5; 1822.498; 1747.022; 1721.154; ; and and
For the graphic illustration, let us consider the initial conditions , the step , and the numbers iterations of time From condition (9) of Theorem 4, the system is permanent if , and One obtains the maximum and minimum values of , , and for The trivial equilibrium points of the model are ,
, and
In addition, is the unique interior equilibrium point verifying condition (91)–(93) of Theorem 13. Then, is globally stable. Hence, the illustrating Figure 2 is for the initial time :
(ii) Interpretation. The trajectories of GDP and capital and (the effect of delay) stabilize, respectively, around , , and when is greater than . The border dynamics show that when the orbit arrives in a border plan, it remains and converges towards a corresponding equilibrium point (see Orbit.2 to Orbit.4 of Figure 2). Moreover, the trivial equilibrium points of the model are all unstable. So, the orbits, are moving away from the borders plans and converge towards Hence, is globally stable, which is illustrated by the figure “phase portrait” of Figure 2.
9. Conclusion
The economic basic model of our work was one of dynamics Kaldor at an effective growth rate whose rate of saving is the Holling-II type and investment rate is from Leslie and Gower type modified in dimension two. Taking into account the time required for the saving to ensure its self-financing of all the investments, one obtains a model with a delay of the Kaldor type modified. This model is bounded and admits an attractor unit (set). Thus, under certain conditions, this model with delay is permanent. On the one hand, this permanence appears in the form of stationary growth of the capital stock and the product (stable interior equilibrium point) and, on the other hand, in the form of cyclic growth of the capital and the product (limiting cycle). In other words, this permanence avoids a shortage of the capital stock or the long-term production. In front of some other conditions, the dynamic stability with delay is also global; i.e., it does not depend on the level of the capital stock and the level of the production at the initial time. The consideration of the delay (due to the time for the economy to finance investments) can justify the bifurcation of an economic model of the stationary growth towards cyclic growth (see Theorem 12, (a)(i)). However, this delay in the model can stabilize an initially unstable equilibrium of the system (see Theorem 12, (b)(ii)), which represents a major economic interest since it allows the saving to get rid of risks from the cyclic growth.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest for this paper.