Stability Theory versus Social Stability

Stability Theory versus Social Stability Vladimir R˘asvan ∗ Department of Automatic Control, University of Craiova, A.I.Cuza, 13 Craiova, RO-200585 Romania (e-mail: [email protected]).

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Abstract: Social stability in the basic conditions for a normal economic and social life allowing survival of mankind. This requires a scientific approach of the problem based both on the social practice and on various scientific theories that may be applied provided adequate models do exist. On the other hand, the social processes are of different nature since they are an interaction of economic processes, social classes interaction and political supervision and control. Since stability theory which is , more or less, the only scientific instrument available and it is essentially based on rigorous modeling, it is desirable to apply this theory to social stability studies. The present paper is an overview of some social/economic mathematical models and their stability properties. Keywords: Stability, Walrasian Dynamics, Gradient-like behavior 1. INTRODUCTION Social stability represents the necessary condition for the existence and survival of the society and of the mankind. Exactly as in the case of engineering or physical systems, social stability is a property of preserving the basic parameters with respect to disturbances which, starting from a certain intensity(amplitude), are viewed as threats to the system operation. The society, the social system, has an undeniable complexity bat the challenge is here to discover the components and the underlying structure that unites them. As it will appear further, the component subsystems are acceptably modeled and for them stability may be studied using the standard stability theory. If we start from the idea that the economic structure of the society is exactly its constitutive basis from where the other elements(social structure, politics) are deduced(while in a very indirect may, via a chain or network of determination) then we may enumerate the following, among other: • economic systems, including manufacture, markets, exchanges, capital flows; worth mentioning that, within a globalist approach, we should add here the natural resources and the environment; • the systems of social relations among determined groups of people. If these large systems are viewed as having their specific dynamics modeled e.g. by differential equations, we may ask if the stability problems for such systems - in the may they arise “at the source” - are well served by the stability notions as they are conceived for the corresponding mathematical objects. 2. ELEMENTS OF STABILITY THEORY AS PART OF THE GENERAL DYNAMICS A. The notion of stability is very expressive : “stable” arises from the Latin stabilis meaning “strong”, “at rest”, “durable”, “solid”. It is so obvious that one might be surprised that there exist enough reasons requiring re-defining an obvious notion. Usually we understand by stability of any phenomenon its property to preserve long enough and with sufficient precision those

forms of existence whose loss would signify the phenomenon ceases to be it self. However both the lay people and the scientists will call stable not necessarily the phenomenon but the system where it is observed, while this does not always hold. For instance: the physical bodies, are they stable? The question make sense provided we mean the stuff : the iron cube is stable but the smoky ball is not. The stability theory in not interested in this kind of stability but in stability of the states and “operation” (achieving the functionality). According to a classical definition Malkin (1966) the theory of stability deals with influence of the perturbations (disturbances) on the motion (evolution). In order to make things more clear, let as discuss the previous assertion concerning “the property to preserve long enough and with sufficient precision those forms of existence whose loss would signify the phenomenon ceases to be it self”. It sends to the necessary properties of a mathematical model of dynamics to be validated, as stated by R. Courant Courant (1965): existence, uniqueness and well-posedness. Existence and uniqueness are the pure expression of the scientific determinism: the model ought to have a solution since in reality it does exist; uniqueness means that identical conditions ought to lead to identical results. Without this determinism, the experiments cannot be repeated for self contained outcomes. Well posedness has the significance given by J. Hadamard: small variations of data (initial conditions or parameters) ought not to result in large variations of the trajectories. This property is necessary both for the requirement that the error affected data should not affect the solutions and approximate solutions should be acceptable with respect to “real” i.e. exact ones. Worth mentioning that the so called World Models used in several widely circulated texts (the Reports addressed to the Club of Rome, for instance) do not meet these requirements. Another worth mentioning remark is that, in their mathematical form, these properties make sense on a finite and small enough interval. But we ask for stability the property “to preserve long enough” and for this reason the basic stability notion - the stability in the sense of A. M. Liapunov - is a property valid on arbitrarily large time intervals.

B. The stability theory as elaborated by A. M. Liapunov operates with some specific notions. Among them we remark first the “trajectory dichotomy”: there are basic trajectories which are of interest and have some repetability properties (equilibria, periodic or almost periodic motions) and perturbed trajectories which are the result of various disturbing factors such as weak ˇ field of forces Cetaev (1931), data mismatches a.o. Recall now the basic definitions of the stability in the sense of A. M. Liapunov - at the colloquial level Definition 1. A basic trajectory of a dynamical system is stable if the initial disturbances which are small enough result in perturbed trajectories that are arbitrarily close to the basic one for arbitrarily large time intervals. If these perturbed trajectories approach asymptotically the basic one, the trajectory is called asymptotic. It is interesting to mention here some definitory observations concerning this notion of stability ⋄ while the property of stability belongs to the basic trajectory, its analysis and establishment are done with reference to the perturbed ones; ⋄ the stability in the sense of Liapunov is a stability with respect to the initial conditions; using a reasoning partly due to Liapunov himself, it may be shown how the effect of short term perturbations is incorporated in the disturbance of the initial conditions; ⋄ while the colloquial expression is “system stability”, this is valid only for linear systems; the property is a trajectory one. Since most of the dynamical models for the social and economic phenomena have several equilibria, the completion of the theory of stability for such systems (initially performed for the synchronization systems) is very useful. We give below some basics Definition 2. a) Any constant solution of a dynamical system is called equilibrium. The set of equilibria E is called also stationary set. b) A trajectory is called convergent if it approaches asymptotically some equilibrium; it is called quasi-convergent if it approaches asymptotically the stationary set as a whole. c) A system is called monostable if every bounded trajectory is convergent; it is called quasi-monostable if every bounded trajectory is quasi-convergent. d) A system is called gradient like if every trajectory is convergent; it is called quasi-gradient like if every trajectory is quasi-convergent. To close this discussion, we mention that the most powerful instrument for analyzing and establishing these properties is the second method of Liapunov. Its power arises, among other justification, from the fact that the properties of a (possibly large) system are obtained using a single function that has nevertheless to be determined in each specific case. 3. DYNAMICS AND STABILITY OF THE COMPETITIVE EQUILIBRIUM IN THE WALRASIAN ECONOMIC MODEL In the entire classical or neo-classical economic school two are the basic paradigms: pure competition (free and unfalsified as the European Constitutional project defines it) and stability of the equilibria. In fact most of the Nobel prizes for Economics

were awarded for theoretical results proving that in various economic models, which are now recognized as being of competitive type, there exist achievable stable equilibria. A. We shall therefore consider such a model, going back to a classic, L´eon Walras; our source is the renown book of Nikaido (1968) and the economic significance may be found there; here we shall give a short self contained review of the mathematical results from Halanay and R˘asvan (1993). The state variables of the model are the prices of the commodities pi ≥ 0, i = 1, . . . , N ; it is not excluded that some prices are zero which means that the corresponding commodities are free. The set where all prices take their values is defined by P = {p ∈ RN : pi ≥ 0 , ∀i ; pj > 0 , j ∈ M} ⊂ RN (1) where M is a given nonempty subset of {1, . . . , N } corresponding to the commodities that cannot be free. Evolution of the prices is described by the law of offer and demand: to describe the action of this law, an excess demand function E : P 7→ RN is considered; for a given system of prices p (the state variables), the components Ei (p) represents the excess of demand with respect to offer for the commodity i. The evolution of the prices is described by the system of differential equations dpi = λi Ei (p) , λi > 0 , i = 1, . . . , N dt

(2)

The qualitative aspect of the law of demand and offer corresponds to the fact that Ei (p) > 0 implies that pi increases and Ei (p) < 0 implies that pi decreases. For this model the main problems are the existence of a system of prices corresponding to equilibrium (i.e. a vector p such that Ei (p) = 0, ∀i) as well as the asymptotic stability of such equilibria. To answer these problems one has to assume some properties of the excess demand function, having natural economic interpretations. These properties are listed below (1) Ei : P 7→ R are homogeneous of degree 0 i.e. Ei (τ p) = Ei (p) for all τ > 0; the significance of this assumption is the independence of the excess demand function of the price scale. (2) Ei (p) are subject to the Walras law hp , E(p)i = 0 i.e. PN 1 pi Ei (p) = 0 for all p ∈ P. (3) The functions Ei (p) are continuously differentiable and subject to the gross substitutability condition: if a price of one commodity grows, the excess demand for the other commodities grows too ∂Ei ≥ 0 , i 6= j , ∀p ∈ P (3) ∂pj This means that the Jacobian matrix of E with respect to the vector p has nonnegative off diagonal elements. (4) There exists some real γ such that Ei (p) ≥ γ, ∀i, ∀p ∈ P. (5) There exists some pˆ ∈ P, pˆi > 0, ∀i such that E(ˆ p) = 0. As an example, consider the case M = {1, . . . , N } and Ei (p) =

N 1 X (aij pj − aji ) , aij ≥ 0 pi 1

Here the fulfilment of (1) and (3) is obvious; also N X 1

pi Ei (p) =

N X N X 1

1

aij pj −

N X N X 1

1

aji pi = 0

(4)

P PN and (2) holds. Property (4) holds for γ = − N 1 aij and 1 existence of the equilibrium means that the linear system N N X X aij pj − pi aji = 0 , i = 1, . . . , N (5) 1

1

has a solution with strictly positive components. B. We reproduce here some properties of the model (2) 1) The function F : P 7→ R+ defined by 1 X p2i 2 1 λi N

F (p) =

(6)

is a global integral of (2). 2) The set P is flow invariant and the existence of the global integral F shows global boundedness of all solutions of (2). 3) All equilibria are stable and every solution tends to an equilibrium (the system is gradient like). 4) The set of equilibria is convex and every equilibrium which is isolated on its level set for F is asymptotically stable on this level set. To end this subject let us mention that there exist similar models in other fields (chemical kinetics, neural networks, biology) where also a single defining function occurs and similar dynamical properties may be pointed out R˘asvan (1998) 4. OTHER CONSIDERATIONS CONCERNING THE MODELS A. We would like to mention here an almost forgotten survey Krotov (1967) published in an almost uncirculated book where there are presented some interesting ideas in the modeling of economic and social phenomena. Typically for the period, the paper starts from the notion of state as basic and states that “the nature laws defining the system are the basic state laws of the system”. The author uses the formalism of the Mechanics with its basic notions such as force and potential. The so called basic law is stated as a variational principle, very much alike to the Statics principle of the minimum of the potential energy (the Lagrange Dirichlet principle of stability for Lagrangean systems). Indeed, if we follow the author construction (op. cit.), it is stated that the entire information about the system is incorporated in the potential. This potential is defined as the potential function of some abstract field of forces F (x). The “effective state” (i.e. the equilibrium) x ¯ differs from any other state by the fact that it ensures a minimum of the potential. It is not difficult to see here the gradient like systems of Halanay and R˘asvan (1993), R˘asvan (1998). The interesting elements of the paper are thus the construction of the forces and their potential. The approach is again taken from Mechanics: the forces are defined using the balancing of the mechanical systems through weights and wires; however the entire mechanical underlining has been removed. Along with standard physical and engineering systems there are introduced models of economics and human behavior for model groups and societies. It appears as interesting and useful to follow this line.

B. In order to end this presentation, it is interesting to mention here the problem of the time delay within the dynamics models of Economics and social interactions, since in most cases they have de-stabilizing effects. The time delay models were introduced firstly in the dynamics of the business cycle by Kalecki, Tinbergen and Frisch - the last two being Nobel prize recipients - in the 30ies of the XXth century. References to their papers are largely available in the standard books on time delay systems. It is interesting to note the reference Dibeh (2001) since it makes reference to the model of Rudolf Hilferding whose book Hilferding (1910) is very actual in the context of the present crisis. But the time delay models are to be met in such areas as long term planning Faerman (1971), prices/unemployment Cargill and Meyer (1974), bioeconomics and renewable resources Clark (1976), various economic control models Pervozvanskii (1975). And, to end, the efforts of the Society for Economic Dynamics are of perennial interest. REFERENCES T. F. Cargill, and R. A. Meyer. Wages, prices and unemployment distributed lag estimates. J. Amer. Statist. Assoc. 69: 98-107, 1974. ˇ N. G. Cetaev. About the stable trajectories of the Dynamics (in Russian). In Stability of motion. Works on Analytical Mechanics pages 245–249, USSR Acad. Publ. House Moscow 1962. C. W. Clark. Mathematical bioeconomics: the optimal management of renewable resources. J. Wiley New York 1976. R. Courant. Equations aux d´eriv´ees partielles hyperboliques et applications. In E. Beckenbach editor Math´ematiques modernes pour l’ing´enieur 1 pages 97–115, Dunod Paris 1965. G. Dibeh. Time Delays and Business Cycles: Hilferding’s model revisited. Review of Political Economy 13:329-341, 2001. E. Yu. Faerman. Problems of long term planning (in Russian). Nauka Moscow 1971. R. Hilferding. Finance Capital: A Study of the Latest Phase of Capitalist Development. Routledge & Kegan Paul London 1981. A. Halanay, and Vl. R˘asvan. Applications of Liapunov Methods in Stability. Kluwer Academic Publishers Dordrecht 1993. V. F. Krotov. Variational principles of the mathematical “Naturphilosophie” (in Russian) In A. M. Letov editor Optimal control systems pages 163–219, Nauka Moscow 1967. I. G. Malkin. Stability of motion (in Russian). Nauka Moscow 1966. H. Nikaido. Convex structures and economic theory. Academic Press London 1968. A. A. Pervozvanskii. Mathematical models of manufacture control (in Russian). Nauka Moscow 1975. Vl. R˘asvan. Dynamical Systems with Several Equilibria and Natural Liapunov Functions. Archivum mathematicum 34: 207-215, 1998.