STUDY OF THE DYNAMICS OF PRODUCT SALES PROCESS WITH THE HELP OF ZOLOTAS MODEL

. A new mathematical model describing the dynamics of sales of goods on the market has been proposed. This model takes into account the following characteristics: the average welfare of buyers, the maximum welfare of buyers, the number of buyers who know about the incoming product and the level of market saturation with this product. Examples demonstrating the features of the constructed model are given.


Introduction
Due to the widespread dissemination of information technologies in mathematical modeling, new directions have emerged that make it possible to more accurately study economic processes [1,4,5,7,8,11].At the same time, models that have long been known in some areas of economics have received new applications in other areas.The transformation of one such well-known model into another economic model is precisely the subject of research in this article.
Normal social security of society in any country allows us to predict the future of this country and take actions leading to an increase in well-being.The famous Greek economist X. Zolotas [10] proceeds from the hypothesis that increasing the production of a large number of goods does not necessarily lead to an improvement in the quality of life.In order to show how social welfare changes, he considers two types of factors that act with greater or lesser relative intensity depending on the already achieved social state.Some factors stimulate, while others restrain the growth of welfare.It is on the opposition of such factors that the Zolotas model is built.
Let us briefly recall the mathematical formulation of the Zolotas model.We denote the level of social welfare of society through V , and through A the critical value of this welfare.Then the restraining factor will be (A − V ), and the stimulating factor will be λV .With this approach, the dynamics of social welfare (the Zolotas model) is determined by the equation where x > 0 is the per capita approach; λ > 0 and A > 0.
The solution of equation (1.1) can be represented as where A ≥ V (x 0 ) > 0 is the initial value [9].Investigating the solution (1.2) of equation (1.1), Zalotas identifies three stages of development of society: I. Society of need; II.Society for Continuous Improvement; III.Society of Crime.
At the present stage of economic development of industrial countries, the restraining factor (I or III) is comparatively stronger than factor II.As a result, the time required for society to rise from the already achieved very high level of well-being to the maximum possible (factor II) is gradually increasing.Reaching the critical value of welfare Zolotas connects both the form of distribution of income and wealth, the degree and speed of natural resources, and the degree of environmental pollution.If we trace the level of social welfare V (x(t)) and the function x(t) (in (1.2) x = x(t) is a function of time), then from solution (1.2) we can determine the indicators λA and V (x 0 ).Then it will be possible to consider the stages of development of the welfare of a particular state and determine the stage in which it is currently located, as well as the distance separating it from the critical point A. (The point A = A * is called critical if there exists a moment t * such that A * = V (x(t * )).It is clear that the point A * can be reached only if The Zolotas model is the basis of the next economic model, the construction of which we are starting.
Let us consider the process of selling some goods about which at time t, out of the number of potential buyers N , only P 3 < N buyers know about these goods.To speed up the sale of products, an advertising campaign was launched.Let us take into account that after advertising, the rate of change in the number of people who know about the product is proportional to both the number of potential buyers P 3 who already know about the product and the number of buyers (N − P 3 ) who do not yet know about it.A mathematical model for describing the number of buyers P 3 (t) who know about the product, provided that time is counted from the moment the advertisement is released, can be written as follows: where K, M ∈ R is the coefficient of proportionality.(Note that if λ = K and M = 1, then equation (1.3) in its structure completely coincides with equation (1.1).) Let us introduce the following notation: µ 0 = −KN and M = −ϵN .Then equation (1.3) can be written as The purchasing power of a particular individual changes depending on the level of critical welfare of people P 2 according to the law: λP 2 , λ > 0.Then, taking into account this law, equation (1.4) can be rewritten as follows Let P 1 be the average level of social welfare, and P 2 its critical level.Then the value P 2 − P 1 is a restraining factor, and −ϵµ 0 P 3 is a stimulating factor.In this case the dynamics of the level of social welfare is determined by a differential equation of the form: Now, if in equation (1.6) we also take into account the market saturation factor λP 0 , then we arrive at the equation In turn, the amount of market saturation with this product P 0 , taking into account the well-being of people P 1 and the number of people P 3 who know about the product, looks like this dP 0 (t) dt = −λP 0 (t) + µ 0 P 1 (t) + ϵµ 0 P 3 (t)P 1 (t). (1.8) Finally, the equation for the critical welfare of people P 2 , taking into account the number of goods on the market, the general average welfare of people P 1 and taking into account advertising P 3 , can be written as follows: Thus, we have obtained a closed system (1.8), (1.7), (1.9), and (1.5) of four differential equations that describe the dynamics of sales of a certain product on the market.A mathematical study of this system will be carried out in the following sections.
Comment.The resulting equations contain four types of variables: P 0 and P 3 are functions describing the numbers of certain groups of buyers; P 1 and P 2 are functions describing the welfare of buyers.Naturally, the parameters included in the equations (1.8), (1.7), (1.9), and (1.5) must be selected so that all variables P 0 , . . ., P 3 would be of the same measurement.In what follows, we will assume that the variables P 0 , . . ., P 3 describe the numbers of different groups of buyers.In this case, the variable P 1 means the number of people with average welfare, and the variable P 2 means the number of people with critical welfare.(From here it follows that the parameter λ = ψ(N ) is a function of N .)

Mathematical preliminaries
We formalize the variables that are included in the system of equations (1.8), (1.7), (1.9), and (1.5): P 0 is the quantity of goods presented on the market (it is assumed that one person can buy only one product; in other words P 0 ≤ N ); P 1 is the number of people in the market (P 1 ≤ N ) with average welfare; P 2 is the number of people (critical level P 2 ≤ N and P 1 → P 2 ), whose welfare has reached the limit of enrichment opportunities in this group of buyers; P 3 are buyers who know that a given product is on the market (P 3 ≤ N ).
We will assume that C = 1.Then, instead of variables P i , we will deal with relative variables P i /(4N ).To simplify further presentation, we will again use the notation P i for relative variables; i = 0, . . ., 3.
Without loss of generality we can assume that P 0 (0) + • • • + P 3 (0) = 1.From here it follows that P 3 = 1 − P 0 − P 1 − P 2 .Using the last relation we reduce the order of system (2.1) by one.Then we have where and We introduce the vector norm: Theorem 2.1.Assume that the conditions of Lemma 2.2 are satisfied.Then system (2.3) has one stable equilibrium point in the hypercube Proof.We need to show that all solutions of system (2.3) starting from the hypercube N 3 are stable.In addition, these solutions are attracted to a single equilibrium point, which is located in the specified hypercube.
Let's consider two cases.

The case ϵ = 0
Taking into account the last remark, we get is (2.3) the line system where Let us show that in this case the matrix A 1 | ϵ=0 is Gurwitz [6].
Obviously, the unique equilibrium point of system (2.4) is determined from the following system of equations: The solution of linear algebraic equations system (2.5) looks like this .
According to the well-known theorem of polynomial algebra (Gua's theorem), if condition a 2 k ≤ a k−1 a k+1 , k = 1, 2, is satisfied for at least one triple of coefficients (a 0 , a 1 , a 2 or a 1 , a 2 , a 3 ), then the polynomial δ 3 + a 1 δ 2 + a 2 δ + a 3 has at least one pair of complex roots.We have Consequently, Gua's theorem does not hold for any of the triples of coefficients of the polynomial.From here it follows that all roots of polynomial δ 3 +a 1 δ 2 +a 2 δ+a 3 are real and negative.Thus, there is a neighborhood of the equilibrium point P 1e ∈ N 3 in which system (2.4) is stable.This equilibrium point is unique in N 3 .
2.2.The case ϵ ̸ = 0 (b1) According to Lemma 2.1, all solutions of system (2.4) are bounded.Note that to prove this fact it was not required that the matrix A 1 be Hurwitz.
Therefore, now we will consider the opposite case: the matrix A 1 is not Hurwitz.
Let's transform equation (2.3) into the following integral equation From here it follows that By virtue of condition (2.2) we get Now let's use the known result of matrix analysis: ∥ exp(A 1 t)∥ ≤ β exp(αt), β > 0 is valid; here α > 0 is the maximum real part of eigenvalues of the matrix A 1 .
Thus, inequality (2.9) can be rewritten in the form Obviously, for t → ∞ for any solution P 1 (t) of equation ( 2.3) we get ∥P 1 (t)∥ → ∞.Thus, if α > 0, then for sufficiently large t > 0 the solution P 1 (t) ̸ ∈ N 3 .Therefore, in order that P 1 (t) ∈ N 3 the matrix A 1 must be Hurwitz.
(b2) We calculate the coefficients of the characteristic polynomial of the matrix A 1 : Now let's use the Hurwitz criterion (2.7).Since all coefficients a 1 , a 2 , a 3 are positive, we only need to check one condition: a 1 a 2 − a 3 > 0. We have If λ ≥ 0, µ 0 ≥ 0, λ + µ 0 > 0, and ϵ ≥ 0, then the last inequality is always true.Therefore, the matrix A 1 (and for ϵ ̸ = 0) is indeed Hurwitz.
(b3) It remains to show that the stable equilibrium point P 1e ∈ N 3 is unique.Let's put Along with the integral equation (2.8), we will consider the same integral equation but with different initial data H i (0) ∈ N 3 , i = 0, . . ., 3.
Suppose that in the hypercube N 3 there are two different equilibrium points: P 1e and H 1e .Then there is a surface G ⊂ N 3 dividing the hypercube into two regions N 3 1 and Thus, we have P 1e ∈ N 3  1 and H 1e ∈ N 3 2 .We estimate the magnitudes lim t→∞ ∥P 1 (t)∥ and lim t→∞ ∥H 1 (t)∥.For this purpose we will use equations (2.8) and (2.11): We introduce the designations and Now we use the notation adopted in inequality (2.10).Then we get (Here, we take into account the fact that the real part of any eigenvalue of the Hurwitz matrix is negative.Therefore, we have −α < 0.) Thus, we have Let us assume that at the moment t = 0 the initial conditions for solutions P 1 (t) and H 1 (t) are taken at some point L on the boundary G.In other words P 1L (0) = H 1L (0).
We introduce a function V (P 1 ) on the set N 3 such that ∀P 1 ∈ N 3 we get Vt (P 1 ) ≤ 0 and ∀L ∈ G we also have Vt (L) = 0. (The derivative of function V (P 1 ) is defined by virtue of system (2.3):Vt (P 1 ) = (∂V (P 1 )/∂P 1 ) • Ṗ1 (t).) Obviously, if the point L ∈ G, then at this point the matrix A 1 is not Hurwitz.(Along the surface G we have either Vt (G) = 0 or Vt (G) doesn't exist.)We obtain a contradiction with the assumption that at any points of the hypercube N 3 the matrix A 1 must be Hurwitz.Therefore, the following relations must be true: G = ∅, N

Generalization
Theorem 2.1 can be generalized to the n-dimensional case, where n > 3. Let's consider a more complex situation than that described in model (2.1).We will assume that the variables P 0 (t) and P n (t) (if n = 3, then P n (t) = P 3 (t)) have the same meaning as in system (2.1).As for the variables P 1 (t) and P 2 (t), we will now assume that there are m > 1 groups, each of which consists of two variables: (P 1 , P 2 ), (P 3 , P 4 ), . . ., (P 2m−1 = P n−2 , P 2m = P n−1 ).(From here it follows that n = 2m + 1.) Note that the economic meaning of the variables P 2i−1 and P 2i is the same as the variables P 1 and P 2 in system (2.1); i = 1, . . ., m.The difference (in the new sales model) is in structuring buyers with various welfare, which was carried out more carefully than it was previously.n+1) .

Let's introduce the following matrix
By symbols P = (P 0 , P 1 , P 2 , . . ., P n ) T ∈ R n+1 and Q = (P 1 , P 2 − P 1 , P 3 − P 2 , . . ., P n−1 − P n−2 , P n − P n−1 , −P n ) T ∈ R n+1 , we denote vectors of R n+1 .Consider the system of differential equations with initial values P(0) ∈ R n .Let's find the sum Σ of all rows of matrix A. Then we get Σ = 0.This means that matrix A has one zero eigenvalue.From here it follows that the characteristic polynomial of matrix A should look like this Here I n ∈ R (n+1)×(n+1) is the identity matrix and ϕ(δ) is a polynomial of degree n.
Then system (3.1) has one stable equilibrium point P e ∈ N n+1 and lim t→∞ P(t) = P e .
Proof.Let us first assume that conditions (a1) and (a2) are satisfied.Since each row of the matrix A contains no more than three elements, the procedure for proving the statements of Lemmas 2.1 and 2.2 can be carried over verbatim to the n-dimensional case.
As to condition (a3), its role in the proof of the theorem will be demonstrated below.
Let's represent the vector Q in the following form where D ∈ R (n+1)×(n+1) .Now we will find the equilibrium points of system (3.1).To do this, it is necessary to solve the system of equations (It is clear that system (3.2) has a trivial solution P = 0.) We sum up all the equations of system (3.1).Then we get Ṗ0 (t) + Ṗ1 (t) + • • • + Ṗn (t) = 0. Integrating the last relation we find In what follows we assume that C = 1.
Thus, we need to solve system (3.2), in which the last equation will be replaced by equation By sequentially solving the equations of system (3.2) we arrive at the following relations Now we use relation (3.3).Then we obtain the following polynomial equation for P n : All coefficients λ, ϵ, and µ 0 in equation (3.5) (except the last one) are positive.Therefore, the number of sign changes for polynomial (3.5) is equal to one.Then, according to Descartes' well-known theorem, equation (3.5) has a single positive root P * n > 0. Now you can use formulas (3.4) to calculate all the remaining coordinates of the equilibrium point.All these coordinates P * 0 , P * 1 ,. . .,P * n−1 are positive and satisfy condition (3.3).
Suppose that the polynomial ϕ(δ) is not Hurwitz.Then, as shown in the proof of Theorem 2.1, the trajectory of system (2.3) leaves the hypercube N 3 .The exclusion of such a situation is the main reason for introducing condition (a3).The proof of Theorem 3.1 is complete.□ Note that an important difference between Theorem 3.1 and Theorem 2.1 is that, unlike Theorem 2.1, we cannot derive condition (a3) from conditions (a1) and (a2).Therefore, in Theorem 3.1 condition (a3) was simply postulated.

Changing assumption (2.2)
Models (2.1) and (3.1), described in the previous sections, were based on one important circumstance: under the conditions of Lemmas 2.1 and 2.2, system (2.1) (and (3.1)) had only one equilibrium point.In this case, it is impossible to perform a complete analysis of the dynamics of the sales market.
Besides condition (4.1), the following condition must also be taken into account.
Taking into account conditions (4.1) and (4.2) leads to the fact that, in contrast to condition (2.2), the behavior of the functions P 0 (t), . . ., P 3 (t) becomes completely different.The difference is that the fulfillment of condition (4.2) imposes additional restrictions on the parameters λ, µ 0 , ϵ in addition to those restrictions that are specified in Lemmas 2.1 and 4.1.In this work we will not write out these mathematical restrictions, but will limit ourselves only to some economic aspects.
The following graphs Fig. 2. The number of sales increases with increasing market saturation and increasing average welfare (despite the decrease in advertising product) (see Fig. 4.2).
3. The number of sales increases with the growth of welfare and the volume of advertising product (see Fig. 4.3).
If the conditions (2.2) are violated, but the conditions of the lemma 4.1 and conditions (4.2) are preserved, several equilibrium points may arise in system (2.1) (see Fig.

Informal behavior of trajectories of system (2.1)
All results obtained in the previous sections have practical significance.However, there always remains a scientific interest in studying the behavior of trajectories of system (2.1), which goes beyond the scope prescribed by Theorem 2.1.This interest is dictated by the following question: are there such values of the parameters of system (2.1), at which the appearance of periodic trajectories (see [2,3]) in the system is possible?
As shown in the proof of Theorem 2.1, if its conditions are satisfied, periodic trajectories in system (2.1) do not exist.However, when changing the conditions of Theorem 2.1, periodic trajectories may appear.Numerical experiments have shown that if parameters λ, µ 0 , and ϵ satisfy the constraints λ > 0, λ + µ 0 < 0, ϵ < 0, and the parameter |ϵ| is small enough, then limit cycles can be born in system (2.1) (see Fig. 5.1 and Fig. 5.

2).
The statements of the last paragraph can be formulated more strictly in the form of the following theorem.
Theorem 5.1.Assume that for system (2.1) the following conditions: (b1) λ > 0, λ + µ 0 ≤ 0, and ϵ < 0; (b3) the matrix A has a pair of complex conjugate eigenvalues with a positive real part, as well as one positive and one zero eigenvalues; (b3) P 3 (0) > 0 are valid.Then in system (2.1) for sufficiently small |ϵ| > 0 there exists a limit cycle.Proof.In what follows we will use equation (1.1) and its solution (1.2).In order that this use to be correct it is necessary to introduce the following redesignations of the parameters of equation (1.1): λ • A → −µ 0 > 0, λ → ϵ • µ 0 > 0. In addition, it is necessary to do replacements x → t and V (x) → P 3 (t).

Conclusion
Model (2.1) showed that, under the conditions of Theorem 2.1, variables P 0 , . . ., P 3 reach their equilibrium values over time.(Mathematically, this means that the equilibrium point of system (2.1) is asymptotically stable [6,9].)The only question is how much this result corresponds to the real dynamics of sales?Note that our task was a mathematical study of solutions to systems (2.1) (with restrictions (2.2) or restrictions (4.1), (4.2)) and systems (3.1).Therefore, we do not in any way pretend to derive important economic conclusions.We only hope that, perhaps, some of our results will be useful to economists who specialize in accelerating the dynamics of market trade.

4 . 1 ,
Fig.4.2, and Fig.4.3 show the change in the number of sales of a product depending on the initial conditions.(The parameters defining the structure of model (2.1), are taken equal to λ = 3, µ 0 = 3, and ϵ = 0.5; conditions (4.2) are valid.)The visual analysis of graphs Fig.4.1, Fig.4.2, and Fig.4.3 allows us to draw the following conclusions: 1.The number of sales increases with increasing market saturation and intensifying advertising campaign (despite the decrease in average welfare) (see Fig.4.1).