COMPUTER MODELLING Computer Modelling & New Technologies, 2000, Volume 4, No.1, 33-40 Transport and Telecommunication Institute, Lomonosov Str.1, Riga, LV-1019, Latvia

PRINCIPLES OF FIELD THEORIES IN MACROECONOMICS YU. SHUNIN, N. DANILOVA The correlated evolution modelling of a total internal product of group of the countries is considered. The features of interaction of their economies, and also condition of stable development are investigated. It is very important to show an economic map of the world in dynamics, in view of the various factors of relationships of the countries and economic regions.

1. Generality of the theoretical description of processes in the nature and society 1.1. "PHYSICAL ECONOMICS" OF LYNDON LAROUSHE The physical economy studies features and principles of development of a ground of material (physical) production with the purpose of quantitative and qualitative improving of filling "of a market basket "on the basis continuous scientifically technological progress providing a long-lived survival of mankind on the Earth [3]. The becoming and development of physical economy affects the society, usual in particular stratums, selfish concerns and comes across a bitter resistance not only supporters of diverse scientific concepts, but also the powers that be. Lyndon LaRoushe1, working out a concept of a physical economy, proves an insolvency as nonMalthusian2 concepts, and so-called "zero growth" theories and suspension scientifically technological progress. He justifies connections between scientifically technological progress, economics of a human labour, increase of its productivity, growth of number and increase of a density of population per unit area [3]. LaRoushe - investigator considers an economic science from items of the philosopher, mathematician, physicist and historian. Just this scale of professional qualities is the key to a success, the way to a comprehension of the laws of a society development. He pays a large attention to social and moral problems, considers not only problems of inducing of a participation of different categories of the workers in a production, but also a possibility of an increasing of living standards and life time in conditions of magnification of a population at the continuous scientifically technological progress and amplification of domination of the person above the Nature. LaRoushe polemically speaks: " How it so? We already lived under the slogan of domination of the person above the Nature. What from this has quitted? Continuous ecological disasters and even catastrophes. Therefore it is necessary not to dominate above the nature, not to take from it, but it is more to give it. ... All business that fairly marring the nature, we suddenly thought: it is impossible to do this! But instead of a searching of normal, ecologically safety methods of domination of the person above Nature, without what the existence of the people is impossible, the pseudo-scientific concepts invoking instead of development a civilizations to not return almost for wildness - in caves, as if for the sake of the Nature saving, have appeared" [3]. In the book of Laroushe [3] is spoken, that such approach not only does not troubleshoot, but drives it in a lockup. The domination of the person above the Nature was, is and will be. Otherwise, a person as the consumer of natural resources for the sake of his survival should vanish or to be dissolved in the nature. Actually, the person will grow higher crops, to apply the more and more perfect engineering, technologies, chemical fertilisers etc. The fertility of soil will be continuously increased. The person will prolong the 1 Lyndon LaRoushe - the American economist and politician, creator of new branch of a science - physical economy, based on a wide circle of the qualitative scientific primary sources, and also works of the Russian scientists (D.I. Mendeleyev, S.Yu. Vitte, V.I. Vernadsky etc)). LaRoushe's economic concept is the universal guidebook on economy, philosophy, history and a number of natural sciences. 2 Malthus Thomas Robert. 1766-1834, English economist. He propounded his population theory in An Essay on the Principle of Population (1798)

33

COMPUTER MODELLING domination above the nature. Unique, that he should not do is not to break, not to pollute and roughly not to delete the Nature. Just with it the applied ecology should struggle. But what is the sense of speculation on domination or subordination of the person to the nature? Thus, the person was, is and will be simultaneous by the creator and user of the nature for the sake of the life on the Earth. The definition of physical economics as a science can be presented in the form of two competing theses [6]. 1 view. The physical economics are an alternate scientific approach, which one preaches technological progress and state regulation of a national economy. And also it is in open opposition to the modern economic doctrine and in every possible ways criticises an existing economic system. 2 view. The physical economy is a science, which its supporters attempt to present as the most progressive, capable ideologically to become the basis of new Revival. But thus the accent is done that the new Revival will come after full corrupting of an old economic system. Only physical economy will ensure the future of civilisation process [6]. 1.2. A HARMONY BETWEEN THE NATURE AND SOCIETY Physics with the help of a mathematical means describes regularities detected experimentally in natural phenomena, or creates hypotheses requiring for check by experience. The mathematical economy not unsuccessfully uses different branches of mathematics for legible formulation of the terms of economy and search between them of quantitative and qualitative ratios. However, a practice shows, that some physical and economic processes and the phenomena have identical mathematical statements. In these cases they speak about analogies different under the nature of problems [2]. In some cases the targeted comparison of problems of physics and economy can suggest outcome of technique and solution of last ones. The analogies of problems can be only formal, and behind them it is nothing, except for a generality of mathematical statements. There are also analogies basing on some fundamental processes, intrinsic both lifeless Nature, and human society, e.g., the thermo-dynamical analogies of market models. In their basement is the process of redistribution: a redistribution of energy and momentums of impacting molecules (physics), and a redistribution of the goods and money between the market participants (economy). Thus corresponding conservation laws are executed [2]. 2. Laws of natural sciences in economy There are four basic principles, which it is worth to consider with the reference to problems of economy: • Hamilton's equation; • Principle Le-Chatelier; • Principle of statistical equilibrium; • Thermo-dynamical analogies. All these concepts are connected with generalised conservation laws. 2.1. HAMILTON'S EQUATION ∂H k = ∂p

,

p = −

∂H , ∂k

(1)

where H (k, p, t) is the Hamilton's function, k is the generalised coordinate, p is the generalised momentum. The set of equations is one of the basic research techniques of systems in an analytical mechanics. 2.2. PRINCIPLE LE-CHATELIER

(πˆ − π )( ˆy − y ) ≥ 0 ,

(2)

34

COMPUTER MODELLING

where (π , у ) and (πˆ , ˆy ) are the couple of effective points, in which the profit reaches a strict maximum on y at appropriate parameter values π. The value of Le-Chatelier principle for the economic analysis is, that it allows to forecast a direction of a modification of parameters of a system owing to change of the environmental conditions. 2.3. STATISTICAL EQUILIBRIUM

( )

s m ψ s g s x* = 0 , dPi + åψ s dg i = 0 ,

dx

s =1

(3)

dx

where g s (x ) ≥ 0 are the resource limitations, s = 1,...,m are the loiter resources during a production, P(x) appropriate to the plan x incomes, x* is the effective in a connection point, ψ s is some scalar factor, x is the production plan. Points satisfying to a set of equations: ψ s ≥ 0 , ψ s g s (x ) = 0 , i = 1,..., n , s = 1,..., m . The system contains n + m of the equations for determination of components of an equilibrium point and m factors values. Selecting from the retrieved set of solutions those, which correspond mentioned conditions, we shall receive all points, which one will really determine final equilibrium positions. 2.4. THERMODYNAMICAL ANALOGIES S i ( x ) → max is the utility function. At limitation on the capital:

m

åπ j x j = Ki , j =1

x j ≥ 0 ,, , where i are

the economic agents, m are the kinds of production marked by the index j, р is the row sector of the prices of products in the market. Actually an expression of a utility function can mean, that the agent i maximises the profit, if for it the products are production factors (e.g., a raw material), or really evaluates usefulness of consumption of these products (e.g., by an amount of calories in them). 3. Interpretation of the field theory concepts in economy

Any mathematical field theories are created on the basis of scalar and of vector field of parameters. We speak here about spatially temporary distributions of values and parameters. The most general approach to problems of universe, including natural sciences and society, is based on the analysis and application of socalled conservation laws, which one or another can be represented in an analytical form. The equation ∂f interpreting some conservation laws or qualities is the known equation of a continuity: + divj = 0 , or ∂t ∂f ± + divj = Σ ± , where f is the local density of the analysable parameter, j is the flux of the parameter, Σ is ∂t the intensity of sources of a field. The mentioned local density in essence is a density of sources of a field, in sense of the field theory. Such concept allows to unite within the framework of one analytical approach the description of the properly non-similar phenomena and processes. Apparently, it is as well that that the conservation laws act equally effectively as in natural sciences (for example, law of conservation of mass, energy conservation law), and, say, in economy (any financial balance).

3.1. FUNDAMENTAL EQUATIONS OF A FIELD THEORY A key for a field theory is the possibility of introduction of a potential Φ , which one being a scalar field is the peculiar generator of vector fields and vector fluxes of parameters. Major outcomes of a mathematical field theory are as a consequence the Laplace equation: ∂ 2Φ ∂y 2

+

∂ 2Φ ∂x 2

=0 ,

(4)

Poisson's equation:

35

COMPUTER MODELLING ∂ 2Φ ∂y

2

+

∂ 2Φ ∂x 2

= f (x , y ) ,

(5)

and the diffusion equation: ∂Φ + div j = f (x , y , t ) . (6) ∂t The next problem of the theoretical analysis is the correct interpretation of potentials and fluxes with reference to definite problems. In particular, in our case this means to problems of macroeconomics. So the generalised potential Φ it is possible to treat, for example, as a gross domestic product3 (GDP) per capita of the population, f(x, y, t) as a generalised functional of resource sources, e.g., as the concentration of capital per square in economical region, and j, e.g., as a generalised flux of investments.

33,946

30,720

27,337

24,956

21,393

1,410

22,394

23,947

3.2. MACROECONOMIC MODEL OF CHANGES OF GDP DISTRIBUTION Consider a group of countries with more and less developed economics to analyse a comparative picture of their economic development. Figure 1 shows a model world economic map of a for group of countries (the wealth of industrially developed countries in a money scale) [1].

1,009

Figure 1. GDP distribution on economic regions (expressed in $).

GDP is influenced by a set of the factors (production of mineral wealths, development of power resources, production of the industrial and agricultural goods, and also food production). For each country it expresses with miscellaneous economic resources. The high rates of country economic growth directly depend on the start level of GDP, internal economic activities and investment fluxes. 4. Simulation of the correlated evolution of GDP of countries group The changes of GDP for the group of countries with a different level of economic development are considered. Figure 2. Spatial relief of model GDP map.

4.1. FORMULATION The general view of group of a set of equations of i objects of simulation has a following kind:

x i = f i (x1 ,..., x n ) ,

(5)

where f i (x1 ,..., x n ) are the influence functions. Let's consider an example, based on a system of 9 equations ( i.e., 9 countries are considered: USA, Japan, Canada, Germany, France, Great Britain, Italy, Russia and Latvia; indexes from 1 to 9, correspondingly; see Figures 2, 3) [1]. Each country has the own specific level of development and own GDP. Rewrite the system of equations (5), expanding the right sides of any one into a set including the second order contributions:

3 Gross Domestic Product - the total value of all goods and services produced domestically by a nation during a year. It is equivalent to gross national product minus net investment incomes from foreign nations. Abbrev.: GDP

36

COMPUTER MODELLING ì dΦ 1 ï dt ï ï dΦ 2 ï dt ï ï dΦ 3 ï dt ï dΦ 4 ï ï dt ïï d Φ 5 í dt ï ï dΦ 6 ï dt ï ï dΦ 7 ï dt ï ï dΦ 8 ï dt ï ï dΦ 9 ïî dt

= − α 1Φ 1 + β 1Φ 12 − γ 1 (Φ 1 − Φ = − α 2Φ

2

+ β 2Φ

= − α 3Φ 3 + β 3Φ = − α 4Φ

4

+ β 4Φ

2 2 2 3 2 4

+γ −γ +γ

= − α 5 Φ 5 + β 5 Φ 52 − γ

2

2

) + δ 1Φ 2

(Φ 1 − Φ 2 ) − δ 2Φ t

(Φ 2

−Φ

(Φ 3

−Φ

5

(Φ 4

− Φ 5 ) + δ 5Φ

3

4

3

) + δ 3Φ 2

4

) − δ 4Φ t 4

= − α 6Φ

6

+ β 6Φ

2 6

+γ

6

(Φ 5

−Φ

6

) − δ 6Φ t

= −α 7Φ

7

+ β 7Φ

2 7

−γ

7

(Φ 6

−Φ

7

) + δ 7Φ 6

= − α 8Φ 8 + β 8Φ 82 + γ 8 (Φ = − α 9 Φ 9 + β 9 Φ 92 − γ

9

7

(Φ 8

,

(6)

− Φ 8 ) − δ 8Φ t − Φ 9 ) + δ 9Φ 8

where α 1 ,..., α 9 are the factors of a degradation of economies (i.e., behaviour of GDP in absence of reproduction), Φ1 ,...,Φ 9 are GDP of given countries, β1 ,..., β 9 are the breeding ratios,

γ 1 ,...,γ 9 are the

influence coefficients (i.e. the transfer parts to the partner country), δ 1 ,...,δ 9 are the parts on loans from the incomes. GDP

GDP per capita

USA Japan

Germany France

Great Britain Italy

Canada

Russia Latvia

33.946 $ 30.720 $22.394 $27.337 $24.956 $23.947 $21.393 $1.410 $ 1.009 $ Figure 3. GDP distribution of group of countries.

4.2. RESULTS AND DISCUSSION The initial data values appropriate to considered countries are selected. At a choice of a degradation factor it is necessary to take into account the level of economic development of any country studied. If it is a developed country, a factor should be selected lesser, if developing one, that, accordingly, it is must be more. It is necessary to take into account a breeding ratio: it is lesser for developed countries and more for

37

COMPUTER MODELLING

developing ones. In our model we suppose various contributions in DGP balances (payments on time, investments, incomes). Thus, on the basement of the above-mentioned explanations we can present a problem numerically: ì x1 = −0.01 ⋅ x1 + 0.5 ⋅ x12 − 0.3 ⋅ (x1 − x 2 ) + 0.05 ⋅ x 2 ï ï x 2 = −0.015 ⋅ x 2 + 0.4 ⋅ x 22 + 0.1 ⋅ (x1 − x 2 ) − 0.01 ⋅ t ï 2 ï x 3 = −0.1 ⋅ x3 + 0.22 ⋅ x3 − 0.2 ⋅ (x 2 − x3 ) + 0.05 ⋅ x 2 ï 2 ï x 4 = −0.02 ⋅ x 4 + 0.3 ⋅ x 4 + 0.05 ⋅ (x3 − x 4 ) − 0.02 ⋅ t ï 2 (7) í x5 = −0.025 ⋅ x5 + 0.27 ⋅ x5 − 0.1 ⋅ (x 4 − x 5 ) + 0.07 ⋅ x 4 , ï 2 ï x 6 = −0.03 ⋅ x 6 + 0.25 ⋅ x 6 + 0.3 ⋅ (x 5 − x 6 ) − 0.03 ⋅ t ï x = −0.12 ⋅ x + 0.20 ⋅ x 2 − 0.15 ⋅ (x − x ) + 0.04 ⋅ x 7 7 6 7 6 ï 7 ï x = −0.35 ⋅ x + 0.09 ⋅ x 2 + 0.2 ⋅ (x − x ) − 0.01 ⋅ t 8 8 7 8 ï 8 2 ï x 9 = −0.7 ⋅ x9 + 0.05 ⋅ x 9 − 0.4 ⋅ (x8 − x 9 ) + 0.09 ⋅ x8 î

where x1 ,..., x 9 are the GPD of considered countries (is researched in a time dependence). As a result of simulation a series of the schedules presented on Figures 4-6 are obtained. Of course, numerical factors in model equation system (7) are conventional. The main tendency, which we point out in this model conditions is the next: countries with developed economy save the rate of growth and countries with non-developed economy behave differently - regressively or progressively.

Figure 4. Model of developed country GDP evolution

Figure 5. Model of typical non-developed country GPD evolution.

38

COMPUTER MODELLING

We shall observe a pulsing picture of world GDP level. The world economic map will be represented rather non-uniform. The level of GDP as integrated parameter of economy allows to designate countries potential investors. Countries with a high DGP per capita should be considered as the most probable donors in the world economy. Just from these countries investment fluxes are more probable. Therefore, countries with the developed economy promote also a development of countries with the non-developed economy. It is worth to point out, that not any country is successfully developed even and with the help of a partner. See model relative GDP evolutions of USA-Latvia and USA-Japan, Figures 6-7. This model results must not be considered as prognostic. But they demonstrate abilities of model approach. We also may carry out numerical analysis of various schemes of macroeconomic relations, their effectiveness and human justice.

Figure 6. Relative model GDP evolution for Latvia and USA.

Figure 7. Relative model GDP evolution for USA and Japan

5. Conclusion Lyndon Laroushe in the framework of a concept physical economics, justified a connection between scientifically technological progress, economies of a human labour, increase of its productivity, growth of number and increase of a density of population per unit area. The physical economics as a science is considered from two sides. Physics with the help of mathematical means describes regularities detected experimentally in natural phenomena, or creates hypotheses requiring for check by experience. The

39

COMPUTER MODELLING

mathematical economy uses different branches of mathematics for formulation of the terms of economy and search between them of quantitative and quality parities. There are analogies basing on some fundamental processes, intrinsic both lifeless nature, and human society. A modelling of GDP distribution evolutions on the basement of a "kinetic" equations gives an universal approach to the analysis of a system macroeconomic relations, allows in case of taking into account realistic factors and laws to "imagine" the "economic" map of the world. References [1] [2] [3] [4] [5] [6] [7] [8]

Digest (LV), 20th of April 2000 Abramov A.P., Ivanilov Yu.P. (1991). Physics and mathematical economy. Moscow. 31p (in Russian) LaRoushe L. (1994) The Science of Physical Economy as the Platonic Epistemological Basis for All Branches of Human Knowledge,” Executive Intelligence Review, Vol. 21, №9-11 (www.e2000.kyiv.org) Neiman E. (1999) Operations on the world financial markets. Europrimex Corp. (www.finansy.ru) Tikhonov A.N., Samarsky A.N. (1977) Equations of mathematical physics. Nauka, Moscow. 735p (in Russian) Blagodeteleva-Vovk S. (1999), e2000.kyiv.org (in Russian) Boldyreva B.G. (1990). Finances of capitalism. Moscow. 383p. (in Russian) McConnell K.R., Brue S.L. (1993) Economics. Moscow. 167p

Received on the 3rd of July 2000

40

PRINCIPLES OF FIELD THEORIES IN MACROECONOMICS YU. SHUNIN, N. DANILOVA The correlated evolution modelling of a total internal product of group of the countries is considered. The features of interaction of their economies, and also condition of stable development are investigated. It is very important to show an economic map of the world in dynamics, in view of the various factors of relationships of the countries and economic regions.

1. Generality of the theoretical description of processes in the nature and society 1.1. "PHYSICAL ECONOMICS" OF LYNDON LAROUSHE The physical economy studies features and principles of development of a ground of material (physical) production with the purpose of quantitative and qualitative improving of filling "of a market basket "on the basis continuous scientifically technological progress providing a long-lived survival of mankind on the Earth [3]. The becoming and development of physical economy affects the society, usual in particular stratums, selfish concerns and comes across a bitter resistance not only supporters of diverse scientific concepts, but also the powers that be. Lyndon LaRoushe1, working out a concept of a physical economy, proves an insolvency as nonMalthusian2 concepts, and so-called "zero growth" theories and suspension scientifically technological progress. He justifies connections between scientifically technological progress, economics of a human labour, increase of its productivity, growth of number and increase of a density of population per unit area [3]. LaRoushe - investigator considers an economic science from items of the philosopher, mathematician, physicist and historian. Just this scale of professional qualities is the key to a success, the way to a comprehension of the laws of a society development. He pays a large attention to social and moral problems, considers not only problems of inducing of a participation of different categories of the workers in a production, but also a possibility of an increasing of living standards and life time in conditions of magnification of a population at the continuous scientifically technological progress and amplification of domination of the person above the Nature. LaRoushe polemically speaks: " How it so? We already lived under the slogan of domination of the person above the Nature. What from this has quitted? Continuous ecological disasters and even catastrophes. Therefore it is necessary not to dominate above the nature, not to take from it, but it is more to give it. ... All business that fairly marring the nature, we suddenly thought: it is impossible to do this! But instead of a searching of normal, ecologically safety methods of domination of the person above Nature, without what the existence of the people is impossible, the pseudo-scientific concepts invoking instead of development a civilizations to not return almost for wildness - in caves, as if for the sake of the Nature saving, have appeared" [3]. In the book of Laroushe [3] is spoken, that such approach not only does not troubleshoot, but drives it in a lockup. The domination of the person above the Nature was, is and will be. Otherwise, a person as the consumer of natural resources for the sake of his survival should vanish or to be dissolved in the nature. Actually, the person will grow higher crops, to apply the more and more perfect engineering, technologies, chemical fertilisers etc. The fertility of soil will be continuously increased. The person will prolong the 1 Lyndon LaRoushe - the American economist and politician, creator of new branch of a science - physical economy, based on a wide circle of the qualitative scientific primary sources, and also works of the Russian scientists (D.I. Mendeleyev, S.Yu. Vitte, V.I. Vernadsky etc)). LaRoushe's economic concept is the universal guidebook on economy, philosophy, history and a number of natural sciences. 2 Malthus Thomas Robert. 1766-1834, English economist. He propounded his population theory in An Essay on the Principle of Population (1798)

33

COMPUTER MODELLING domination above the nature. Unique, that he should not do is not to break, not to pollute and roughly not to delete the Nature. Just with it the applied ecology should struggle. But what is the sense of speculation on domination or subordination of the person to the nature? Thus, the person was, is and will be simultaneous by the creator and user of the nature for the sake of the life on the Earth. The definition of physical economics as a science can be presented in the form of two competing theses [6]. 1 view. The physical economics are an alternate scientific approach, which one preaches technological progress and state regulation of a national economy. And also it is in open opposition to the modern economic doctrine and in every possible ways criticises an existing economic system. 2 view. The physical economy is a science, which its supporters attempt to present as the most progressive, capable ideologically to become the basis of new Revival. But thus the accent is done that the new Revival will come after full corrupting of an old economic system. Only physical economy will ensure the future of civilisation process [6]. 1.2. A HARMONY BETWEEN THE NATURE AND SOCIETY Physics with the help of a mathematical means describes regularities detected experimentally in natural phenomena, or creates hypotheses requiring for check by experience. The mathematical economy not unsuccessfully uses different branches of mathematics for legible formulation of the terms of economy and search between them of quantitative and qualitative ratios. However, a practice shows, that some physical and economic processes and the phenomena have identical mathematical statements. In these cases they speak about analogies different under the nature of problems [2]. In some cases the targeted comparison of problems of physics and economy can suggest outcome of technique and solution of last ones. The analogies of problems can be only formal, and behind them it is nothing, except for a generality of mathematical statements. There are also analogies basing on some fundamental processes, intrinsic both lifeless Nature, and human society, e.g., the thermo-dynamical analogies of market models. In their basement is the process of redistribution: a redistribution of energy and momentums of impacting molecules (physics), and a redistribution of the goods and money between the market participants (economy). Thus corresponding conservation laws are executed [2]. 2. Laws of natural sciences in economy There are four basic principles, which it is worth to consider with the reference to problems of economy: • Hamilton's equation; • Principle Le-Chatelier; • Principle of statistical equilibrium; • Thermo-dynamical analogies. All these concepts are connected with generalised conservation laws. 2.1. HAMILTON'S EQUATION ∂H k = ∂p

,

p = −

∂H , ∂k

(1)

where H (k, p, t) is the Hamilton's function, k is the generalised coordinate, p is the generalised momentum. The set of equations is one of the basic research techniques of systems in an analytical mechanics. 2.2. PRINCIPLE LE-CHATELIER

(πˆ − π )( ˆy − y ) ≥ 0 ,

(2)

34

COMPUTER MODELLING

where (π , у ) and (πˆ , ˆy ) are the couple of effective points, in which the profit reaches a strict maximum on y at appropriate parameter values π. The value of Le-Chatelier principle for the economic analysis is, that it allows to forecast a direction of a modification of parameters of a system owing to change of the environmental conditions. 2.3. STATISTICAL EQUILIBRIUM

( )

s m ψ s g s x* = 0 , dPi + åψ s dg i = 0 ,

dx

s =1

(3)

dx

where g s (x ) ≥ 0 are the resource limitations, s = 1,...,m are the loiter resources during a production, P(x) appropriate to the plan x incomes, x* is the effective in a connection point, ψ s is some scalar factor, x is the production plan. Points satisfying to a set of equations: ψ s ≥ 0 , ψ s g s (x ) = 0 , i = 1,..., n , s = 1,..., m . The system contains n + m of the equations for determination of components of an equilibrium point and m factors values. Selecting from the retrieved set of solutions those, which correspond mentioned conditions, we shall receive all points, which one will really determine final equilibrium positions. 2.4. THERMODYNAMICAL ANALOGIES S i ( x ) → max is the utility function. At limitation on the capital:

m

åπ j x j = Ki , j =1

x j ≥ 0 ,, , where i are

the economic agents, m are the kinds of production marked by the index j, р is the row sector of the prices of products in the market. Actually an expression of a utility function can mean, that the agent i maximises the profit, if for it the products are production factors (e.g., a raw material), or really evaluates usefulness of consumption of these products (e.g., by an amount of calories in them). 3. Interpretation of the field theory concepts in economy

Any mathematical field theories are created on the basis of scalar and of vector field of parameters. We speak here about spatially temporary distributions of values and parameters. The most general approach to problems of universe, including natural sciences and society, is based on the analysis and application of socalled conservation laws, which one or another can be represented in an analytical form. The equation ∂f interpreting some conservation laws or qualities is the known equation of a continuity: + divj = 0 , or ∂t ∂f ± + divj = Σ ± , where f is the local density of the analysable parameter, j is the flux of the parameter, Σ is ∂t the intensity of sources of a field. The mentioned local density in essence is a density of sources of a field, in sense of the field theory. Such concept allows to unite within the framework of one analytical approach the description of the properly non-similar phenomena and processes. Apparently, it is as well that that the conservation laws act equally effectively as in natural sciences (for example, law of conservation of mass, energy conservation law), and, say, in economy (any financial balance).

3.1. FUNDAMENTAL EQUATIONS OF A FIELD THEORY A key for a field theory is the possibility of introduction of a potential Φ , which one being a scalar field is the peculiar generator of vector fields and vector fluxes of parameters. Major outcomes of a mathematical field theory are as a consequence the Laplace equation: ∂ 2Φ ∂y 2

+

∂ 2Φ ∂x 2

=0 ,

(4)

Poisson's equation:

35

COMPUTER MODELLING ∂ 2Φ ∂y

2

+

∂ 2Φ ∂x 2

= f (x , y ) ,

(5)

and the diffusion equation: ∂Φ + div j = f (x , y , t ) . (6) ∂t The next problem of the theoretical analysis is the correct interpretation of potentials and fluxes with reference to definite problems. In particular, in our case this means to problems of macroeconomics. So the generalised potential Φ it is possible to treat, for example, as a gross domestic product3 (GDP) per capita of the population, f(x, y, t) as a generalised functional of resource sources, e.g., as the concentration of capital per square in economical region, and j, e.g., as a generalised flux of investments.

33,946

30,720

27,337

24,956

21,393

1,410

22,394

23,947

3.2. MACROECONOMIC MODEL OF CHANGES OF GDP DISTRIBUTION Consider a group of countries with more and less developed economics to analyse a comparative picture of their economic development. Figure 1 shows a model world economic map of a for group of countries (the wealth of industrially developed countries in a money scale) [1].

1,009

Figure 1. GDP distribution on economic regions (expressed in $).

GDP is influenced by a set of the factors (production of mineral wealths, development of power resources, production of the industrial and agricultural goods, and also food production). For each country it expresses with miscellaneous economic resources. The high rates of country economic growth directly depend on the start level of GDP, internal economic activities and investment fluxes. 4. Simulation of the correlated evolution of GDP of countries group The changes of GDP for the group of countries with a different level of economic development are considered. Figure 2. Spatial relief of model GDP map.

4.1. FORMULATION The general view of group of a set of equations of i objects of simulation has a following kind:

x i = f i (x1 ,..., x n ) ,

(5)

where f i (x1 ,..., x n ) are the influence functions. Let's consider an example, based on a system of 9 equations ( i.e., 9 countries are considered: USA, Japan, Canada, Germany, France, Great Britain, Italy, Russia and Latvia; indexes from 1 to 9, correspondingly; see Figures 2, 3) [1]. Each country has the own specific level of development and own GDP. Rewrite the system of equations (5), expanding the right sides of any one into a set including the second order contributions:

3 Gross Domestic Product - the total value of all goods and services produced domestically by a nation during a year. It is equivalent to gross national product minus net investment incomes from foreign nations. Abbrev.: GDP

36

COMPUTER MODELLING ì dΦ 1 ï dt ï ï dΦ 2 ï dt ï ï dΦ 3 ï dt ï dΦ 4 ï ï dt ïï d Φ 5 í dt ï ï dΦ 6 ï dt ï ï dΦ 7 ï dt ï ï dΦ 8 ï dt ï ï dΦ 9 ïî dt

= − α 1Φ 1 + β 1Φ 12 − γ 1 (Φ 1 − Φ = − α 2Φ

2

+ β 2Φ

= − α 3Φ 3 + β 3Φ = − α 4Φ

4

+ β 4Φ

2 2 2 3 2 4

+γ −γ +γ

= − α 5 Φ 5 + β 5 Φ 52 − γ

2

2

) + δ 1Φ 2

(Φ 1 − Φ 2 ) − δ 2Φ t

(Φ 2

−Φ

(Φ 3

−Φ

5

(Φ 4

− Φ 5 ) + δ 5Φ

3

4

3

) + δ 3Φ 2

4

) − δ 4Φ t 4

= − α 6Φ

6

+ β 6Φ

2 6

+γ

6

(Φ 5

−Φ

6

) − δ 6Φ t

= −α 7Φ

7

+ β 7Φ

2 7

−γ

7

(Φ 6

−Φ

7

) + δ 7Φ 6

= − α 8Φ 8 + β 8Φ 82 + γ 8 (Φ = − α 9 Φ 9 + β 9 Φ 92 − γ

9

7

(Φ 8

,

(6)

− Φ 8 ) − δ 8Φ t − Φ 9 ) + δ 9Φ 8

where α 1 ,..., α 9 are the factors of a degradation of economies (i.e., behaviour of GDP in absence of reproduction), Φ1 ,...,Φ 9 are GDP of given countries, β1 ,..., β 9 are the breeding ratios,

γ 1 ,...,γ 9 are the

influence coefficients (i.e. the transfer parts to the partner country), δ 1 ,...,δ 9 are the parts on loans from the incomes. GDP

GDP per capita

USA Japan

Germany France

Great Britain Italy

Canada

Russia Latvia

33.946 $ 30.720 $22.394 $27.337 $24.956 $23.947 $21.393 $1.410 $ 1.009 $ Figure 3. GDP distribution of group of countries.

4.2. RESULTS AND DISCUSSION The initial data values appropriate to considered countries are selected. At a choice of a degradation factor it is necessary to take into account the level of economic development of any country studied. If it is a developed country, a factor should be selected lesser, if developing one, that, accordingly, it is must be more. It is necessary to take into account a breeding ratio: it is lesser for developed countries and more for

37

COMPUTER MODELLING

developing ones. In our model we suppose various contributions in DGP balances (payments on time, investments, incomes). Thus, on the basement of the above-mentioned explanations we can present a problem numerically: ì x1 = −0.01 ⋅ x1 + 0.5 ⋅ x12 − 0.3 ⋅ (x1 − x 2 ) + 0.05 ⋅ x 2 ï ï x 2 = −0.015 ⋅ x 2 + 0.4 ⋅ x 22 + 0.1 ⋅ (x1 − x 2 ) − 0.01 ⋅ t ï 2 ï x 3 = −0.1 ⋅ x3 + 0.22 ⋅ x3 − 0.2 ⋅ (x 2 − x3 ) + 0.05 ⋅ x 2 ï 2 ï x 4 = −0.02 ⋅ x 4 + 0.3 ⋅ x 4 + 0.05 ⋅ (x3 − x 4 ) − 0.02 ⋅ t ï 2 (7) í x5 = −0.025 ⋅ x5 + 0.27 ⋅ x5 − 0.1 ⋅ (x 4 − x 5 ) + 0.07 ⋅ x 4 , ï 2 ï x 6 = −0.03 ⋅ x 6 + 0.25 ⋅ x 6 + 0.3 ⋅ (x 5 − x 6 ) − 0.03 ⋅ t ï x = −0.12 ⋅ x + 0.20 ⋅ x 2 − 0.15 ⋅ (x − x ) + 0.04 ⋅ x 7 7 6 7 6 ï 7 ï x = −0.35 ⋅ x + 0.09 ⋅ x 2 + 0.2 ⋅ (x − x ) − 0.01 ⋅ t 8 8 7 8 ï 8 2 ï x 9 = −0.7 ⋅ x9 + 0.05 ⋅ x 9 − 0.4 ⋅ (x8 − x 9 ) + 0.09 ⋅ x8 î

where x1 ,..., x 9 are the GPD of considered countries (is researched in a time dependence). As a result of simulation a series of the schedules presented on Figures 4-6 are obtained. Of course, numerical factors in model equation system (7) are conventional. The main tendency, which we point out in this model conditions is the next: countries with developed economy save the rate of growth and countries with non-developed economy behave differently - regressively or progressively.

Figure 4. Model of developed country GDP evolution

Figure 5. Model of typical non-developed country GPD evolution.

38

COMPUTER MODELLING

We shall observe a pulsing picture of world GDP level. The world economic map will be represented rather non-uniform. The level of GDP as integrated parameter of economy allows to designate countries potential investors. Countries with a high DGP per capita should be considered as the most probable donors in the world economy. Just from these countries investment fluxes are more probable. Therefore, countries with the developed economy promote also a development of countries with the non-developed economy. It is worth to point out, that not any country is successfully developed even and with the help of a partner. See model relative GDP evolutions of USA-Latvia and USA-Japan, Figures 6-7. This model results must not be considered as prognostic. But they demonstrate abilities of model approach. We also may carry out numerical analysis of various schemes of macroeconomic relations, their effectiveness and human justice.

Figure 6. Relative model GDP evolution for Latvia and USA.

Figure 7. Relative model GDP evolution for USA and Japan

5. Conclusion Lyndon Laroushe in the framework of a concept physical economics, justified a connection between scientifically technological progress, economies of a human labour, increase of its productivity, growth of number and increase of a density of population per unit area. The physical economics as a science is considered from two sides. Physics with the help of mathematical means describes regularities detected experimentally in natural phenomena, or creates hypotheses requiring for check by experience. The

39

COMPUTER MODELLING

mathematical economy uses different branches of mathematics for formulation of the terms of economy and search between them of quantitative and quality parities. There are analogies basing on some fundamental processes, intrinsic both lifeless nature, and human society. A modelling of GDP distribution evolutions on the basement of a "kinetic" equations gives an universal approach to the analysis of a system macroeconomic relations, allows in case of taking into account realistic factors and laws to "imagine" the "economic" map of the world. References [1] [2] [3] [4] [5] [6] [7] [8]

Digest (LV), 20th of April 2000 Abramov A.P., Ivanilov Yu.P. (1991). Physics and mathematical economy. Moscow. 31p (in Russian) LaRoushe L. (1994) The Science of Physical Economy as the Platonic Epistemological Basis for All Branches of Human Knowledge,” Executive Intelligence Review, Vol. 21, №9-11 (www.e2000.kyiv.org) Neiman E. (1999) Operations on the world financial markets. Europrimex Corp. (www.finansy.ru) Tikhonov A.N., Samarsky A.N. (1977) Equations of mathematical physics. Nauka, Moscow. 735p (in Russian) Blagodeteleva-Vovk S. (1999), e2000.kyiv.org (in Russian) Boldyreva B.G. (1990). Finances of capitalism. Moscow. 383p. (in Russian) McConnell K.R., Brue S.L. (1993) Economics. Moscow. 167p

Received on the 3rd of July 2000

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