Cyclic Growth and Global Stability of Economic Dynamics of Kaldor Type in Two Dimensions

Abstract

This article proposes nonlinear economic dynamics continuous in two dimensions of Kaldor type, the saving rate and the investment rate, which are functions of ecological origin verifying the nonwasting properties of the resources and economic assumption of Kaldor. The important results of this study contain the notions of bounded solutions, the existence of an attractive set, local and global stability of equilibrium, the system permanence, and the existence of a limit cycle.

1. Introduction

The nonlinear complex dynamics have been introduced in the analysis of the economic phenomena to explain on the one hand the fluctuations noticed in the study of the chronological series and on the other hand the economic crisis in the capitalist system. The economists Goodwin (1967) and Kaldor (1955-1956) used the dynamic samples to explain that the graphs of cyclic and chaotic evolution are endogeneous to the economic system itself.

To simplify things, a great number of these models have been elaborated with more restrictive assumption such as linearity. The challenges of structural reforms of dynamics justify the fact that mathematicians are interested in them. Our contribution will thus consist in proposing economic models inspired by the ecological models whose lessons may be important in terms of analyzing systems and their regulation in this context of climate protection (cf. [1, 2]).

The basic economic models that we use in this article are those of Kaldor proposed in the works of Hans-Walter Lorenz (cf. [3]).

Our study will consist first of modifying the models of Kaldor conferring them with the ecological properties adapted to economy. Therefore, we propose a dynamic model typical of Kaldor-Holling-2 and Leslie-Gower with some modifications. Next, we will study the qualitative comportment of the model at level 2. We set criteria for which we have on the one hand the marking out of the solutions and the existence of an attractive set and, on the overhand, the local stability of the equilibrium and the permanence of the system.

At last we study the global stability of one interior equilibrium through the construction of the Lyapunov function.

2. Dynamics of Kaldor Ecologic in Two Dimensions

2.1. Economic Dynamics of Type Kaldor with the Effective Growth Rate in Two Dimensions

Let us start by giving the notations and definitions of the rates of parameters and functions in applied economies.(1)The productivity of capital is the quotient of the GDP by the capital . We set that .(2)The rate of investment is the quotient of the investment by the GDP. We note that .(3)The saving rate of the GDP is the quotient of the saving by the GDP. We have .(4)The rate accumulation of the capital is denoted by .(5)We design, respectively, by and the monetary depreciation rate and the monetary adjustment coefficient.(6)The ratio saving-capital will be noted as .

Let us consider the original model of Kaldor (cf. [3] page 44):

In order to establish a connection between the economic models of Kaldor and the ecological patterns, let us give up the coercion of monetary scales by writing investment and saving functions in relation to the investment rate, of capital and saving hoarding. Let us also replace the growth rate of the GDP of model (1) by its effective growth rate (cf. [3, 4]). Therefore we get the following assumption.

Assumption 1 (Kaldor with effective growth). (1) Effective growth with the coefficient of monetary mending, and the tendency, , such that .
(2) The ratio saving-capital, , is a function verifying
(3) The investment rate is a function verifying and there is a threshold so that .
(4) The accumulation rate of the capital is a function verifying ,

With Assumption 1, we get below the dynamics of Kaldor with an effective growth rate whose and can have ecological properties.

Interpretation. The dynamics of Kaldor with an effective growth rate (2) present some similarities to the classical ecological dynamics. The function define the action of the capital upon production . It stimulates the increase of the production when the investment rate is superior to the saving rate and stops it in the end.

The function defines in its part the action of the production over the capital. The economic model alone summarizes two types of ecological interaction such as mutualising type and the prey-predators type. Let us consider the economic assumption of ecological inspirations on investment rate , the rate of capital accumulation , and the ratio saving-capital .

Assumption 2. where

Obviously, the investments are first funded by capitals. We can thus suppose that . Next, the investors adjust the rate in relation to the realities for they hate to invest in vain. However it is not necessary to invest more when net profit is beyond expectancies. Let us estimate the losses of the investment by using the function , where is the maximum value of the losses of the investment rate and is the maximum value of the stock of capital. Then we get . For simplicity, we can take , where and are the constants depending on the economic policies of investments. Then, we get a rate of investment that verifies the economic requirements (5) of Assumption 1.

Concerning the accumulation of capital, it is known that so . If then with the part of GDP converted in the stock of capital and . Therefore verifies the economic constraint (11) of Assumption 1.

Concerning the saving, let us take , where is the ratio saving-capital. The function verifies condition (4) of Assumption 1.

Suppose now that the tendency is linear and decreasing: . In fact, the tendency of the growth rate of GDP is at the start, a constant for a given period (forthcoming). But it faces some losses due, for example, to corruption, bribes, slush funds, tax haven, whitening of fraudulent funds, manipulations of accounts and media, and any other harmful activity to the growth of the GDP (cf. [5], pages 11, 15-16, and 35–65). Those losses are estimated to with the maximum value (monetary) of the GDP that we can get from this economy for the given period.

Then, model (2) becomes the economic dynamics: with and . indicates the product and the stock of capital, and and indicate, respectively, the increasing speed of the product and the stock of capital.

System (6) defined in this way is a more realistic system. It takes into account a great deal of economic observations of interactions between the product (GDP) and the stock of capital of an economy; namely,(1)in the absence of the stock of capital, there will not be any explosion of the GDP because the increase of GDP becomes logistic so that, despite the technical progress, the economic production remains limited,(2)in the absence of the production of the GDP, the economy will not be in short of capital if because the evolution is as well logistic due to the diversification of the economy or the opportunities to convert a stock of physical capital in the stock of monetary capital,(3)when the production (GDP) becomes abundant, there is a “saturation” of the ratio saving-capital and of the investment expressing the adoption of a nonwasting policy of the economic resources,(4)when the production of the GDP is insufficient, the ratio saving-capital gets adapted and becomes proportional to the available GDP in order to avoid a shortage of production.

2.2. Presentation of Model (6) Reduced

In order to facilitate the qualitative study of system (6) that possesses parameters , let us change variables by reducing the number of parameters to

Definition of New Variables

Definition of the Parameters System (6) becomes then the reduced system: with and so that

3. Boundness of Model (9) and Existence of a Positively Invariant Attracting Set

In this section, we give the conditions of the boundness of the capital and the stock of capital justifying the fact that the economic resources are limited.

Lemma 3. The interior and the boundary of the positive quadrant are, respectively, unvarying for system (9).

Proof. Given , if , due to the continuity of and over the compact then we have

So if then .

If then and if then , .

Therefore , , and , .

Lemma 4 (cf. [6]). Given and continuous and derivable function so that there is verifying , then, ,

Definition 5 (see [7, 8]). A solution of (9) is said to be a boundary in , if there is compact of and a time () so that , we have for every

Theorem 6. Let us suppose that . Let us set downLet us consider the following set:(1).(2).(3) is unvarying for model (9).(4) is an attractive region for any solution of model (9) from the positive quadrant .

Proof. Let us consider system (9). Let us set down (1)Let us show that . We have and Then, with and Then through the application property (13) of Lemma 4, we have , with . Therefore, , so that , (2)Let us prove that . Let us set down . We have and ; then, we have . Now , . Then . Let us set down and then, . Now ; then, through the application property (12) of Lemma 4, we have , ; then ,   so that , .(3)Let us show that is unvarying for model (9). Given , from Lemma 3, of (18) and (19), , , , and So is unvarying for model (9).(4)Let us show that is an attractive solution of model (9). Let us show that . We deduce from Lemma 3, (18), and (19) that , , and , . Consequently is an attractive region for any solution of model (9) from the positive quadrant .

4. Equilibrium of Model (9)

We are now going to give the conditions of a balanced growth (stationary) of the product and the stock of capital and the quantitative values of the parameters in equilibrium.

4.1. Case in Which

4.1.1. Trivial Equilibrium

Proposition 7. (1) If then system (9) admits two trivial equilibriums:(2) If then system (9) admits three trivial equilibriums:

4.1.2. Interior Equilibrium

Theorem 8. Let with (1)System (9) does not admit the interior equilibrium if .(2)Any interior equilibrium of system (9) satisfies the following relations:

Proof. Let us consider model (9); then, . Given an equilibrium of the model (9),(1)if then so for any , we have ; therefore the system does not admit any interior equilibrium.(2)Given an interior equilibrium, then we have Therefore,  Posingand , we obtain

Corollary 9. Let us suppose that and considering the polynomial defined in Theorem 8, let us set down , , , , , and .
System (9) admits a unique interior equilibrium such that and in each of the following cases:(a), .(b) and one of the following conditions is verified:(i), , and (ii), , and (iii), , , , and .(iv), , , and .(v), , , and .

Proof. Given the polynomial is defined in Theorem 8 and such that and .
Let us pose ; then, the equation will be reduced to with , , . The characteristic equation of is . Let us pose , , . Given, , , solutions of certifying , then, , , , and , the solutions in of , are , , , .
So, the roots in of are for
The roots , , , and in of prove the following system: and
By examining the number of positive roots of and knowing that and because , we get Corollary 9.

4.2. Case in Which

Considering the conditions and , in system (9), we get the following system: with and

4.2.1. Trivial Equilibrium

Proposition 10. (1) If , then, system (28) admits two trivial equilibriums:(2) If , then, system (28) admits three trivial equilibriums:

4.2.2. Interior Equilibrium

Theorem 11. Given that with (1)if then system (28) does not admit any interior equilibriums,(2)if then any interior equilibrium of system (28) verifies the following system:

Corollary 12. Let us suppose that and consider the polynomial of Theorem 11. System (28) admits a unique interior equilibrium so that and if one of the following conditions is verified: (i).(ii), , and .(iii) and .

Proof. If is an interior equilibrium of (28) and , the polynomial, and are stipulated in Theorem 11. Then, and .
We get .
By examining the number of positive roots of in which and , we obtain researched results.

5. Local Stability and Permanence of Model (9)

In this section, we first define the conditions in which this balanced growth of the product and the stock of capital of the economy are stable or unstable. Then, let us examine the possibility of having permanently those two parameters of the economy (sustainable development). This permanence of the product and the stock of capital of the economy are noticed either through the convergence (of both parameters) towards a stable equilibrium or through a fluctuation of both parameters around an unstable equilibrium, that is, the convergence towards a limited cycle.

5.1. Local Stability of Model (9)

In system (9), we pose the following: We have and .

Let us note that , the Jacobian matrix of the system, is linear around . Then, we have with

Theorem 13 (local stability). (1) Stability of :(a) is an unstable node if .(b) is an unstable saddle point if ,(i)repulsive along the direction ,(ii)attractive along the direction .(2) Stability of : (a) is stable if .(b) is an unstable saddle point if ,(i)attractive along the direction ,(ii)repulsive along the direction .(3) Stability of for : (a) is stable if .(b) is an unstable saddle point if , repulsive along the direction and attractive along the direction .(4) Given , an interior equilibrium of (9) verifying system (23) of Theorem 8, and its associate Jacobian matrix, (a) is stable (node or a focus) if and ,(b) is marginal or a center if and ,(c) is unstable if or and ; more precisely,(i) is a node or a focus if and ,(ii) is a unstable saddle if .With the trace and the determinant of , the polynomial , the vectors , and .

Proof. Given , the Jacobian matrix of the system, is linear around the equilibrium for and of , we have the following:(1)Stability of : We have  The numbers and are the eigenvalues of . Eigenspace associated with is , where indicates the vectorial subspace generated by the family: with . The eigenspace associated with is . We have the following:(a)If then and ; then is an unstable node and its unstable manifold is (b)If then and ; then is an unstable saddle point, the unstable manifold of which is and the stable manifold is .(c)If then and ; therefore is an equilibrium the unstable manifold of which is and of central manifold is .(2)Stability of : We have  So and are the eigenvalues of  The eigenspace associated with is . Note that . The eigenspace associated with is  We get the following:(a)If then and ; then is stable and its stable manifold is .(b)If then and ; then is an unstable saddle point of which the stable manifold is and the unstable manifold is .(c)If then and ; consequently is an equilibrium the stable manifold of which is and the central manifold is .(3)Stability of : We have  So and are the eigenvalues of  Let us note that . The eigenspace associated with is . The eigenspace associated with is . We get the following:(a)If then and ; then, is stable and its stable manifold is .(b)If then and ; then is an unstable saddle the unstable manifold of which is and the stable manifold of which is .(4)Stability of : Let and be the eigenvalues of . Thus, let us note its trace and its determinant. The eigenspace associated with is and the eigenspace associated with is .(a)If and then and . So, is stable (stable node or stable focus).(b)If and then Therefore is a marginal or a center.(c)If or and then is unstable. In fact,(i)if and then and or and are conjugated complexes. So, is node or an unstable center,(ii)if then is an unstable saddle.

5.2. Permanence of Model (9)

Definition 14 (see [8]). Given solution of a differential system(1)a component of the solution of (39) is said to be weakly persistent if ,(2)a component of the solution of (39) is said to be highly persistent if ,(3)a component of the solution of (39) is said to be uniformly persistent if there is such that ,(4)System (39) is said to be dissipative as for any component of the solution there is a constant such that ,(5)System (39) is said to be permanent if it is uniformly persistent and dissipative.

Let be complete metric space and for an open set such that . Further, we shall take .

Definition 15 (see [8]). A flow or semiflow on under which and are forward invariant is said to be permanent if it is dissipative and if there is a number such that any trajectory starting in will be at least at a distance from for all sufficiently large .

Definition 16 (see [8]). (1) The -limit set is said to be isolated if it has a covering of pairwise disjoint sets which are isolated and invariant with respect to the flow or the semiflow both on and on .
(2) The set is said to be acyclic if there exists an isolated covering such that no subset of is a cycle.

Lemma 17 (see [8]). Suppose that a semiflow on leaves both and forward invariant, maps bounded sets in to precompact set for , and it is dissipative. If in addition (1) is isolated and acyclic,(2) for all , where is the isolated covering used in the definition of acyclicity of and denotes the stable manifold,then the semiflow is permanent.

Theorem 18. Let us assume that and ; then we pose Let us consider the following assumptions: Under the assumptions and , model (9) is permanent and any positive solution of (9) verifies

Proof. Given and , its frontier, and , we know that and are invariants for model (9) (cf. Lemma 3) and that is attractive bounded for any trajectory from (cf. Theorem 6). Let us assume that and and let us apply Lemma 17: (1)Let us justify the fact that model (9) is dissipative over :(a)Let us show that . Posing and , we have and So, from property (14) of Lemma 4, we have .(b)Let us show that . Posing and , then, we have and We get for .(c)We deduce that model (9) is dissipative over as soon as the assumption is verified.(2)Let us prove that is isolated and acyclic. We have . On the one hand, ; now the stable manifold is if and is unstable if . Then any trajectory from other than approaches if for . On the other hand, ; now the stable manifold of is if and is unstable if ; then any trajectory from other than approaches if and . Given that then we deduce that is isolated and acyclic if (if the assumption is verified).(3)Let us justify that . We have and . If , , and then ; now . So if .Definitively, system (9) is permanent.

5.3. Limit Cycle of Model (9)

The theorem below presents the conditions for a cyclic growth of the product and the stock of capital of the economy.

Theorem 19. Let us recall the notations of Theorem 13. Let us suppose that the assumptions of Theorem 18 are verified and that model (9) admits a unique interior equilibrium . If and then model (9) admits a limit cycle contained in the attractive region .

Proof. Under the assumption of Theorem 18, model (9) is permanent and if and then the unique interior equilibrium is unstable so model (9) admits a limit cycle contained in the compact and bounded region (from Poincaré-Beddicton’s theorem).

6. Global Stability of Model (9)

We now define the conditions in which stability of the product and the stock of capital of the economy are global; that is, they do not depend on the quantities produced and the level of the stock at the initial period. For this study, we define appropriate Lyapunov function.

Theorem 20. Posing and , let us consider the following assumptions:Under assumptions (43)–(46), the unique interior equilibrium of model (9) is globally and asymptotically stable.

Proof. Let us consider system (9). Let us suppose that assumption (44) is verified; then, model (9) admits a unique interior equilibrium .
Let us note that . We have and such thatWe have the Lyapunov function such that .
Then, . Now and , a unique interior equilibrium of (9); then, Therefore,Let us pose the following: Consequently, Then, . Thus, if and , .
Let us determine the conditions on the control parameters of model (9) such that and , (cf. [9], pages 110-111). (1)Let us overestimate . We have . Then . Therefore , if .(2)Let us overestimate . We have . Now ; then, . Consequently, , . Then, . Therefore, , , if (3)Let us deduce that . We know that , , and , . Now , , if and , . Then, , , if assumptions (43)–(46) are verified. Therefore the unique interior equilibrium of model (9) is globally and asymptotically stable if assumptions (43)–(46) are verified.

7. Conclusion

Our work used the Kaldor model as basic economic model. By including it at the level of the investment rate and saving rate compatible ecological functions, we encourage the economic actors to adopt a behaviour permitting very rapidly entering a stability area (attracting set ). This stability can be noticed on the one hand in the form of stationary growth of the stock of capital and the product (stable interior equilibrium) and on the other hand in the form of cyclic growth of the capital and the product (limit cycle). We therefore guarantee, under certain conditions, the permanence of the stock of capital, , and that of the product, , in the economy avoiding, in that way, a shortage of the stock of capital or the production in the long term. Under certain conditions, this stability of the financial system (in relation to the capital and the product) is global; that is, it does depend on the level of the stock of capital and the level of production at the initial period.

In the first consideration, the model can be applied to a state, regional organisation, or to an enterprise. In the case of an enterprise, the product refers to the monetary value of the production. We can then infuse the existing production functions such as Cobb-Douglas, Leontief, and CES. In that case, we can substitute the saving with the quantity of work and the tendency will show a technical progress.

Secondly, the model can also be applied as an ecological model of two species where one of the species (e.g., man) “cultivate” the other species for its survival or to prevent the loss of that species through a culture rate (investment rate) which is nonnull.

Conflicts of Interest

The authors declare that they have no conflicts of interest.