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International Journal of Differential Equations
Volume 2018, Article ID 7212795, 18 pages
https://doi.org/10.1155/2018/7212795
Research Article

Global Stability of an Economic Model with a Continuous Delay of Kaldor Type Modified

UFR de Mathématiques et Informatique Université Félix Houphouët Boigny d’Abidjan Cocody 22 BP 582 Abidjan 22, Côte d’Ivoire

Correspondence should be addressed to Tetchi Albin N’guessan; moc.liamg@ihctetnibla and Sahoua Hypolithe Okou A Kpetihi; moc.liamtoh@ihitepkauoko

Received 7 March 2018; Revised 13 May 2018; Accepted 17 May 2018; Published 2 August 2018

Academic Editor: Dongfang Li

Copyright © 2018 Aka Fulgence Nindjin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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, , < . ThusHence, , > 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 obtainsIf 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

On plan, model becomes

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, , , ifThe 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

On the plan model becomes

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

On the plan model becomes

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 consider model .

Let us pose that , , and for with