Diffuse (field) macroeconomic models, Model of limiting generalization...

Diffuse (field) macroeconomic models

The above macroeconomic models are models of systems with lumped parameters.

Meanwhile, the real economy is a system distributed in the country's space, and even in the global space. Moreover, the distribution of financial, labor, energy and raw materials does not coincide with each other, and the distribution of consumption volumes does not coincide with the distribution of resources, which has to be taken into account in the spatial distribution of production capacities.

The most adequate apparatus for describing and optimizing such spatial collisions within the framework of the macroeconomic model is the apparatus of mathematical field theory, appropriately generalized to the case of a multitude of physically and economically diverse sources of the field of the economy, in contrast to traditionally physically homogeneous sources (charges or masses) in the electromagnetic and gravitational fields.

The model of limiting generalization of values ​​

To begin with, we will consider a diffuse macroeconomic model in which the values ​​are devoid of qualitative certainty and are characterized only by the surface density γ (χ, y) of the territorial distribution (in rubles/m2), where hiu are the coordinates of this point on the surface of the earth. Then, denoting


where h (x, y) - the attractiveness (in bits) of a given place is similar to (7.13), in which p (x, y) is the degree of satisfaction of needs at the same point, according to field theory, we have


i.e. This is the nominal thickness (in meters) of the layer of economic activity, where R 0 - absolute availability (in bits/m ∙ rub); R k - relative availability of values; I ^ R0 = R;


The particular solution (7.31a) represents


where S is the area of ​​integration; r is the distance from a given point to each point of the area of ​​integration.

In the case of economic symmetry, the economic Coulomb or Newton law follows from (7.316)

or (7.31c)

where J is the cost at the point r = 0.

If γ expresses the surface density of value in rubles/m2, ah is the attractiveness of the given place (in bits), then E - vector of intensity of the field of economic interests gets the dimension in bits/m and, characterizing the state of the economy at a given point, like h, , in contrast to the latter, also indicates the direction of economic interests.

In other words, E is the winning vector at a given point from moving to each meter in the E direction.

In transient dynamic economic regimes (7.31a) turns into


where v0 - speed of distribution of economic processes, i.e. the speed of transportation of values ​​at a given point; and its particular solution is the lagging attractiveness of this point in the economic space


In the case of the economic symmetry of the conditions

or (7.32b)

In other words, if at a given point the value of J, changes, then at a distance r from this point the attraction h of the terrain will change only after a time y/s.

The relations considered characterize the spatial and temporal relationship between the attractiveness h of each given point of the economic space and the value of the value of γ at each point of this space.

These relationships allow, on the one hand, to select the most attractive co-ordinates x, y of the place of economic activity by the criterion h => max or by a more universal criterion for the maximum density of the economical energy c = γ/ι = & gt; max, and even optimize the location of economic activity throughout the territory,


and on the other hand, express the parameters of the chain model (see Figure 7.3) through the parameters of the diffuse model


Consider a simple example.

Suppose there are two resource sources J 1 and J 2 at a distance r0 from each other, and we need to find the optimal location of production consuming these resources.

The attractiveness h 1 and h 2, created by Jj and J2, according to (7.31), make = RJ 1 /2nr 1 uI 2 = RJ 2 /2nr 2, where r x is the distance to J 1 and r2 is the distance to J 2, and + r2 = r0.

To solve the problem, of course, one can find the maximum of the aggregate attractiveness h = h 1 + h 2, which determines r r and r2, but it is easier to proceed from the fact that at the optimal point Ej = -E2, i.e. according to (7.31), RJ 1 /2nr = RJ 2 /2nr, whence


The problem becomes more complicated if, for example, the density γ of the work force changes along r0 in accordance with the law (y ^/r0) exp (-r: n0) and (y2z2/r0) exp (-z2/ro). Then we have to look for a maximum not h, and c = y h, which leads to rx = 0.5ro (l + lny, J, /y 2 ^ 2 ^ and r2 = 0.5g0 (1 - InyjJj/y2J2).

In addition, if at some point t Q the resource providers J 2 suddenly stop deliveries, then production at the distance r2 will cease only at the moment t 0 + r 2/v0.

However, there is still a connection between the vector of linear density) turnover of values ​​in a particular point of the economic space and the vector attractiveness of this turnover at other points


where j is measured in R/m ∙ s, and A - in the bit ∙ s/m, and the relationship between j and γ gives the value of the law of conservation of matter


According to this law, the value of the cost expressed at j at a given point is associated with a decrease in its stock at the same point expressed by γ.

The particular solution (7.45) gives the magnitude and direction at each point of interest in turnover at the remaining points


and in the case of the central symmetry of the field


where R 2 /2kv 2 = L a is the local inertia of the economy field, and ll is the local revolution at the center of symmetry of the field, so that A = L 0 I n/g.

In other words, the further the interested legal entity (observer) is from the object of his interests, the less his interest in the turnover of values ​​on this object, although this interest is proportional to the value 1 A of this turnover and its local inertia L0, measured in bits ∙ c2/rub.

In the transitional mode


and its particular solution gives a lagging turnover appeal


In the case of the central symmetry of the field


i.e. the reverse dependence of the attractiveness of turnover on distance is preserved, but with a delay of r/v0.

These relationships allow, on the one hand, to choose the most attractive by turnover values ​​of the place on the economic map or by the criterion A => max, or by the criterion of the density of the energy-kinetic energy c; = A-j = & gt; max, or, finally, to optimize the distribution of turnover throughout the territory

For example, if, as in the previous example, it is necessary to choose the location of the production of values ​​between two points with consumption turns 1 r and/2, the distance between which is r 0, then according to (7.356), the vector attractiveness Aj = L 0 1 /i x and A2 = L 0 I 2/r2, and the energy supply reserve densities, respectively, are 0 ', γ,/r0) exp (-rj/r0) and (j 2 r 2/r0) exp (- r2/r0) - so that the density of the total economic energy

The maximum value of c is reached when r x = 0.5ro (l + In /, j x /I 2 j 2) and r2 = 0.5ro (l -In 1 X/2; 2), and, of course, r < > x + r 2 = r 0.

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