Economic reality creates problems that can only be solved with the help of a new paradigm for describing the economic system. Classical mathematical economics is based on the concept of rational consumer choice, which is generated by a certain preference relation on a certain set of goods desired by the consumer, and the concept of maximizing the firm’s profit. The meaning of the concept of rational consumer choice is that it is determined by a certain utility function, which determines consumer’s choice by maximizing it on a certain budgetary set of goods. In this case, the choice of consumers is independent (see: [1] - [5] ).

In reality, choices of consumers are not independent because they depend on the firms supply. In addition to the supply of firms, consumer choice is also determined by the information about the state of the economic system that the consumer has and evaluates at the time of choice. In turn, the supply of firms is formed on the basis of consumer needs and purchasing power.

By information about the state of the economic system we mean certain information about the equilibrium vector of prices and production processes that are implemented in the economic system at the equilibrium price vector.

The new concept of the description of the economy system is to construct the stochastic model of the economy system based on the principle that the firms supply is primary and the consumers choice is secondary. Consumers make their choice having information about the state of the economy system that is taken into account by them under the choice. Firms make decisions relative to productive processes on the basis of information about the real needs of consumers and this principle is called the agreement of the supply structure with the choice structure.

To construct such a theory, it is necessary to formulate adequate to the reality the theory of consumers choice and decisions making by firms. Thus, the main foundations of the stochastic model of economy system are the notions of consumers choice and decisions making by firms relative to productive processes. Under uncertainty these two notions have the stochastic nature. Besides, the theory of consumers choice must take into account the structure of supply and mutual dependence of the consumers choice.

So, the sense of the new paradigm of the description of the economic phenomena is to construct models of the economy systems, describing adequately consumers choice and decisions making by firms relative to productive processes, that give possibility to use them for finding the conditions of the stable growth of real economy systems (see: [6] - [8] ).

In the works [9] - [12] , the ideas of [8] are implemented to build the concept of sustainable development at the micro-economic level.

This article is a direct continuation of the works of [9] - [12] for the case of an aggregate description of the economy. The aim of the paper is to identify the conditions under which the sustainable development of an economic system described in aggregate terms takes place. This problem is solved in the case of a non-aggregate description of the economic system in papers [9] [10] . The main hypothesis is the assumption that prices for goods produced in an economic system are formed under the influence of market forces of supply and demand and the taxation system. The same works formulate what the principle of sustainable development means in mathematical form, which takes into account market forces and the taxation system that affect the formation of prices for goods produced in the economic system. This work is original in terms of both the problem statement and the obtained results. Statement of the problem to describe all taxation systems that ensure sustainable development was formulated in a general form in works (see [9] - [12] ). It was partially solved in works [11] [12] . In this work, it is solved completely for the “input-output” production model.

The considered problem is significantly different from the existing one in the literature (see, for example [13] [14] ). There, the influence of the taxation system on the economic equilibrium is studied. Moreover, production technologies are described by the Cobb-Douglas model.

All obtained results are original and obtained by the author for the first time. For the first time, it was possible to establish the dependence of the taxation system on production technologies and production volumes. It has been proven that there are no other taxation systems that ensure sustainable development. Among all taxation systems, there are those that are called perfect because they are equitable to producers.

This paper aims to develop a methodology for studying economic systems to find out how far a real economic system deviates from the sustainable development regime defined in the paper. The latter is important for detecting economic systems in order to identify whether the economic system is moving towards recession [15] - [17] .

Sustainable development for different countries at the macroeconomic level was studied in works [18] - [21] . Based on the constructed concept of describing economic systems [8] , the paper [22] builds a model of international trade and studies the trade of the G20 countries, and the paper [23] studies the trade of the 8 largest economies in the world.

Section 2 contains axioms of aggregated description of economic systems. It means that all productive process in the economy can be divided into the set of elementary productive processes. Such description of economy we call not aggregated one. Every elementary productive process consists from expenditure vector to produce one type of good and output vector of that type of good. The basic assumption about the elementary production process is that there is a linear relationship between the output vector and the input vector. The latter leads to the fact that any output vector under a non-aggregated description can be presented using the so-called Leontiev matrix of direct costs. The main assumption relative to the direct cost matrix is its productivity.

Proposition 1 shows that every productive matrix has a spectral radius less than unity. Due to the fact that the number of elementary production processes is on the order of 10⁹, a non-aggregated description of the economy is impossible. In practice, elementary production processes are aggregated into a small number of so-called pure industries. To come from a non-aggregated description to an aggregated one, a certain mapping of the set of all elementary production processes into a smaller set of pure industries is used. Now the aggregate direct cost matrix depends in a non-linear way on the vector of gross output in the non-aggregate description and the vector of prices that is formed in the production process when contracts for the supply of inputs or intermediate inputs or final goods are concluded. We introduce the important concept of the relative price vector and establish a relationship for aggregate values in the presence of a relative price vector.

Proposition 2 establishes important relations in which the relative vector of prices appears. Proposition 3 states that if the non-aggregated direct cost matrix is productive, then the aggregated direct cost matrix will be as well. It is established that for the aggregated values there are balance ratios similar to the non-aggregated balance ratios. The introduced relative price vector will play an important role in the description of taxation systems for the aggregate description of the economy. In Proposition 5, an important formula for aggregated added value in the presence of a vector of relative prices is proved. Sustainable development was defined as the production of goods and services that leads to the complete clearing of markets in a given period of economic functioning under the influence of the market mechanism of pricing goods and services (see for example [10] ).

The market mechanism for establishing the prices of goods and services is understood as the equilibrium of supply and demand for resources and produced goods in the production process, taking into account the tax system.

When we describe the economy in aggregate, the concept of sustainable development must be clarified. In Section 3, to clarify what we mean by the sustainable development of the economy, the principle of correspondence is introduced. In order to take into account the influence of the taxation system on pricing in the economic system, the concept of a relative price vector is introduced, which must satisfy, in accordance with the introduced principle of correspondence, a certain system of nonlinear equations. The study of this problem was started in the papers [9] - [12] . In the paper [10] , all taxation systems that ensure sustainable economic development in the input-output production model was described. The Theorem 1 proves the existence of a strictly positive solution of a certain linear system of equations with respect to the price vector, which will satisfy the vector of relative prices for taxation systems, which can transfer the economy into a sustainable development regime.

In the Theorem 2, the formula for aggregate value added is established when the price vector used for aggregation is adjusted for the vector of relative prices.

Among the taxation systems that can transfer an economic system into the mode of sustainable development, perfect taxation systems are distinguished. They are characterized by the fact that the gross value added in an industry is equal to the value of the product created in the same industry.

We describe taxation systems under which the economic system is able to transfer into a subsidy regime. That is, there exists a market pricing mechanism in which there is a strictly positive equilibrium price vector such that under this taxation system certain industries require subsidies to exist.

Theorem 3 provides the necessary and sufficient conditions under which a tax system is able to transfer the economic system into the mode of sustainable development. In fact, it provides the description of tax systems under which an economic system is able to transfer into the mode of sustainable development.

Theorem 4 formulates the necessary and sufficient conditions for taxation systems under which the economic system is able to transfer into the mode of sustainable development.

It states that there are no taxation systems other than those described that are able to transfer an economic system into the mode of sustainable development.

Theorem 5 establishes the conditions for the gross output vector under which the economic system is able to transfer into the mode of sustainable development.

Theorem 6 states if under the taxation system, given by the formula (34), the economic system is able to transfer into the mode of sustainable development, then the set of all industries is divided into two non-intersecting sets in which the added value either does not exceed the value of the product produced or strictly exceeds it.

In the Definition 5, we determine the tax system that is able to transfer the economic system into the mode with subsidies.

Theorem 7 gives sufficient conditions for tax systems under which the economic system is able to transfer to the regime where certain industries require subsidies.

Theorem 8 establishes the existence of perfect tax systems that is able to transfer the economic system into the mode of sustainable development.

Theorem 9 gives the sufficient conditions under which the perfect taxation system is able to transfer the economic system in the mode of sustainable development.

More specifically, provided that the vector of gross outputs is a solution of the system of equations in the “input-output” production model with a positive right-hand side, there is always a perfect tax system given by an explicit formula by which the economic system is able to transfer into the mode of sustainable development. Moreover, the gross value added generated in each industry is strictly positive. It is shown that in the mode of sustainable development under the established taxation system, the gross value added is equal to the value of the product created in this industry.

In real economic systems, not all branches of production have a strictly positive gross value added. There may be several reasons for this: obsolete production technologies, significant imports of consumer goods, raw materials, etc.

Not all tax systems are able to transfer the economic system into the mode of sustainable development. The equilibrium price vector generated by the tax system and production technologies and the demand for resources may lead to the negative value added in certain industries.

The Theorem 10 establishes sufficient conditions under which the taxation system can transfer the economic system into a subsidized regime.

The definition 7 defines when the described aggregate economic systems can function in a sustainable development mode.

Theorem 11 gives a complete description of economic systems that satisfy the Definition 7, and a formula for the taxation system under which sustainable development of the economy takes place.

Finally, Theorem 12 and Theorem 13 formulate the conditions under which the aggregated economic system in relation to the price vector formed in the economic system can function in the mode of sustainable development under a perfect taxation system. In Theorem 12, such a condition is that the gross output vector belongs to the interior of the cone formed by the vectors of the columns of the matrix of total costs under a perfect taxation system.

Theorem 13 states that if the vector of gross output does not belong to the positive cone formed by the vectors columns of the matrix of total costs under a perfect taxation system, there are branches of the economy that require subsidies.

In the following definitions, the important notion of consistency of the taxation system under consideration with the taxation system existing in the real economic system is given.

The Definition 8 introduces the concept of consistency of the taxation system under consideration with the aggregated real taxation system.

The Definition 9 distinguishes among the consistent taxation systems those that can transfer the economic system to a sustainable development regime. The Theorem 14 gives the necessary and sufficient conditions under which an economic system can be transferred to the regime of sustainable economic development by a consistent taxation system.

The Theorem 15 gives sufficient conditions under which the economic system described in aggregate can be transferred to the mode of sustainable development by a consistent taxation system.

The next Section 4 is devoted to the study of the proximity of the US economy to the regime of sustainable development based on Theorem 11, 12, 13, 14, 15. For this purpose, the information about the US economic system presented in the “input-output” tables for the years 2000-2014 is presented in canonical form. The methodology of research of economic systems for the purpose of their stay in the mode of sustainable development is presented. It is shown that the American economy during 2010-2014 was close to the regime of sustainable development.

Let’s assume that $m$ types of goods are produced in the economic system. We assume that all produced goods are ordered and any set of them will be denoted by a non-negative vector of $m$-dimensional subset $R_{+}^m$. In reality, the production of any product can be represented by a production process $\left(x,y\right)$, where the vector $x$ is an input vector and $y$ is an output vector.

Axiom 1. Production of goods and services in the economic system can be presented as a set of elementary production processes $\left(x^i,y^i\right),i=\overline{1,m}$, where $x^i$ is an input vector of the i-th type of good and $y^i={\left\{{\delta }_{ij}{x}_i\right\}}_{j=1}^m$ is a certain output vector of that type of good, ${\delta }_{ij}$ is a Kronecker symbol.

Such description of economy we call not aggregated one.

Axiom 2. There is a linear relationship between the vectors $x^i$ and $y^i$ of productive process $\left(x^i,y^i\right)$: to produce the vector of goods $y^i={\left\{{\delta }_{ij}{x}_i\right\}}_{j=1}^m$ it is necessary to spend the vector of goods $x^i={\left\{a_{ij}{x}_i\right\}}_{j=1}^m,i=\overline{1,m}$, where $a_{ij}\ge 0,i,j=\overline{1,m}$.

As a result of the assumption of linearity of elementary production processes, we obtain: if to introduce into consideration the matrix $A={\left|a_{ij}\right|}_{i,j=1}^m$ of direct costs, then to produce the output vector $y={\left\{x_i\right\}}_{i=1}^m$, it needs to expend the vector $x=Ay.$ The matrix $A$ is known in the literature as the Leontief matrix. The vector $y-x=y-Ay=c$ is called the final consumption vector. Consumer needs are considered primary, that is, the supply is primary, and the demand is secondary, therefore, as a rule, the final consumption vector $c={\left\{c_i\right\}}_{i=1}^m$ is important. If we introduce a re-designation, denoting the output vector by $x={\left\{x_i\right\}}_{i=1}^m$ and the input vector by $Ax$, then the main output equation takes the form

$x-Ax=c.$(1)

Definition 1. A non negative matrix $A$ is called productive if at least for one strictly positive vector $c_0\in R^m$ there exists a strictly positive solution $x_0$ of the system of equations (1).

Axiom 3. The matrix of elementary direct costs $A$ is a productive one.

Proposition 1. If the matrix $A$ is productive, then its spectral radius is strictly less than unity.

Proof. For the vector $x\in R^m$, we introduce a norm by setting $\Vert x\Vert =\underset{k\in M}{\text{max}}\frac{\left|x_k\right|}{x_k^0}$, where $x^0={\left\{x_i^0\right\}}_{i=1}^m$ is a solution of the system of Equation (1) for a certain strictly positive vector $c^0={\left\{c_i^0\right\}}_{i=1}^m\in R^m$, and $M=\left\{1,2,\cdots ,m\right\}$. Then for the norm of the operator $A$ in the space $R^m$ there is the estimate $\Vert A\Vert \le \left(1-\underset{k\in M}{\text{max}}\frac{c_k^0}{x_k^0}\right)<1$. It immediately follows that the system of Equation (1) is solvable for any non-negative vector $c\in R^m$ in the set of non-negative solutions. □

To present information about the economic system as a whole, it is necessary to provide information about the matrix $A$. But more than 10⁹ types of goods are produced in the economic system. This means that the number of elementary production processes has the same order. Therefore, such a presentation is impossible. Below we describe how to present such information with a smaller number of values.

Let’s go to this description. We need to describe the economic system with a smaller number of so-called pure industries. We introduce two sets of integer numbers: $M=\left\{1,2,\cdots ,m\right\}$ and $N=\left\{1,2,\cdots ,n\right\}$ The set $M$ numbers the elementary production processes in the economic system, and the set $N$ numbers the aggregate industries. We assume that $\left|M\right|>\left|N\right|$, where $\left|M\right|=m$ is a number of elements in the set $M$ and $\left|N\right|=n$ is a number of elements in the set $N$. We introduce into consideration a mapping $f$ of the set $M$ onto $N$. The mapping $f$ we call the aggregation map.

In the production process, a strictly positive price vector $p={\left\{p_i\right\}}_{i=1}^m$ is formed between producers who have entered into contracts with each other for the supply of goods. It is formed under the influence of supply and demand for goods in the economic system and under the influence of the formed system of both direct and indirect taxes. By vectors $x={\left\{x_i\right\}}_{i=1}^m$, $p={\left\{p_i\right\}}_{i=1}^m$, and the matrix $A$ we introduce the following aggregated quantities: $X={\left\{X_k\right\}}_{k=1}^n$, $C={\left\{C_k\right\}}_{k=1}^n$, $\bar{A}={\Vert {\bar{a}}_{ki}\Vert }_{k,i=1}^n$, where

$X_k=X_k\left(p,x\right)=\underset{\left\{l,f\left(l\right)=k\right\}}{{\displaystyle \sum }}{p}_l{x}_l,C_k=C_k\left(p,x\right)=\underset{\left\{l,f\left(l\right)=k\right\}}{{\displaystyle \sum }}{p}_l{c}_l,k=\overline{1,n}.$(2)

${\bar{a}}_{ki}={\bar{a}}_{ki}\left(p,x\right)=\frac{{\displaystyle \underset{\left\{s,f\left(s\right)=k\right\}}{\sum }{\displaystyle \underset{\left\{r,f\left(r\right)=i\right\}}{\sum }{p}_s{a}_{sr}{x}_r}}}{{\displaystyle \underset{\left\{l,f\left(l\right)=i\right\}}{\sum }{p}_l{x}_l}},k,i=\overline{1,n}.$(3)

Further, in order to take into account the impact on pricing in the economic system of the taxation system, we introduce the concept of a relative price vector $\hat{p}={\left\{{\hat{p}}_i\right\}}_{i=1}^n$ from $R_{+}^n$. If the strictly positive price vector $p={\left\{p_i\right\}}_{i=1}^m\in R_{+}^m$ is an aggregation vector and $\hat{p}={\left\{{\hat{p}}_i\right\}}_{i=1}^n\in R_{+}^n$ is a strictly positive relative price vector, then the price vector $p\left(\hat{p}\right)={\left\{{\hat{p}}_{f\left(i\right)}{p}_i\right\}}_{i=1}^m$ will be called the corrected price vector aggregation, which takes into account the impact of the taxation system on pricing in the economic system.

Proposition 2. Let $\hat{p}={\left\{{\hat{p}}_i\right\}}_{i=1}^n$ be a certain relative strictly positive vector from $R_{+}^n$ and let $p={\left\{p_i\right\}}_{i=1}^m$ and $x={\left\{x_i\right\}}_{i=1}^m$ be strictly positive vectors of prices and gross outputs from the set $R_{+}^m$, respectively. Then the following identities

${\bar{a}}_{ki}\left(p\left(\hat{p}\right),x\right)=\frac{{\hat{p}}_k{\bar{a}}_{ki}\left(p,x\right)}{{\hat{p}}_i},k,i=\overline{1,n},$(4)

$X_k\left(p\left(\hat{p}\right),x\right)={\hat{p}}_k{X}_k\left(p,x\right),k=\overline{1,n},$(5)

$C_k\left(p\left(\hat{p}\right),x\right)={\hat{p}}_k{C}_k\left(p,x\right),k=\overline{1,n},$(6)

are valid, where $p\left(\hat{p}\right)={\left\{{\hat{p}}_{f\left(i\right)}{p}_i\right\}}_{i=1}^m$.

Proof. The proof of Proposition 2 follows from the definition of aggregated matrix $\bar{A}={\Vert {\bar{a}}_{ki}\Vert }_{k,i=1}^n$, given by the formula (3), the gross output vector $X$ and the consumption vector $C$ given by the formulas (2). □

Strictly positive vector $\hat{p}={\left\{{\hat{p}}_i\right\}}_{i=1}^n$ from the subset $R_{+}^n$ is called the relative price vector.

Proposition 3. Suppose that the matrix $A={\Vert a_{ki}\Vert }_{k,i=1}^m$ is productive one, then the aggregated matrix $\bar{A}={\Vert {\bar{a}}_{ki}\Vert }_{k,i=1}^n$ is also productive.

Proof. Due to the productivity of matrix $A$, for any strictly positive vector $c={\left\{c_i\right\}}_{i=1}^m$ there exists a strictly positive solution $x={\left\{x_i\right\}}_{i=1}^m$ to the systems of Equation (1). Then the aggregated strictly positive vector $X={\left\{X_i\right\}}_{i=1}^n$ is a solution of the system of Equation (8) with the strictly positive vector $C={\left\{C_i\right\}}_{i=1}^n$. The latter means the productivity of the matrix $\bar{A}={\Vert {\bar{a}}_{ki}\Vert }_{k,i=1}^n$.

For the price vector $p={\left\{p_i\right\}}_{i=1}^m$, that was formed under the action of competitive forces that are the result of the taxation system, the production technologies, the monetary policy of central bank, we introduce into consideration the vector $\delta ={\left\{{\delta }_i\right\}}_{i=1}^m$, where

${\delta }_i=p_i-\underset{s=1}{\overset{m}{{\displaystyle \sum }}}{a}_{si}{p}_s,i=\overline{1,m}.$(7)

The vector $\delta ={\left\{{\delta }_i\right\}}_{i=1}^m$ is called the vector of created added values. Under the conditions described above, this vector may not be strictly positive. This means that not all production technologies are capable of creating the positive added values under the competitive conditions. However, as a result of aggregation, it may turn out that the added values aggregated ${\text{Δ}}_j=\underset{\left\{l,f\left(l\right)=j\right\}}{{\displaystyle \sum }}{\delta }_l{x}_l,j=\overline{1,n}$, may be either all strictly positive or not strictly positive in all industries. Since the vector $x={\left\{x_i\right\}}_{i=1}^m$ is a solution of the system of Equation (1), then the vector $X={\left\{X_i\right\}}_{i=1}^n$ satisfies the system of equations

$X_k-\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{ki}{X}_i=C_k,k=\overline{1,n}.$(8)

Due to the relations (7), the following equations

$X_j-\underset{s=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{sj}{X}_j={\text{Δ}}_j,j=\overline{1,n},$(9)

are true, where ${\text{Δ}}_j=\underset{\left\{l,f\left(l\right)=j\right\}}{{\displaystyle \sum }}{\delta }_l{x}_l,j=\overline{1,n}$.

Definition 2. Description of the economy in terms of value by the vector of gross output $X={\left\{X_i\right\}}_{i=1}^n$, by the vector of final consumption $C={\left\{C_i\right\}}_{i=1}^n$, by the vector of added values $\Delta ={\left\{{\Delta }_i\right\}}_{i=1}^n$, and by the matrix $\bar{A}={\Vert a_{ki}\Vert }_{k,i=1}^n$, which satisfy the relations (8) and (9), we are called by aggregated description.

A consequence of the systems of Equations (8) and (9) is the equality

$\underset{k=1}{\overset{n}{{\displaystyle \sum }}}{C}_k=\underset{s=1}{\overset{n}{{\displaystyle \sum }}}{\text{Δ}}_s,$(10)

which means that the sum of the values of final product created in all industries is equal to the sum of added values created in all industries.

Proposition 4. If ${\Delta }_j>0,j=\overline{1,n}$, then the inequalities $\underset{s=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{sj}<1,j=\overline{1,n}$, are true.

Proof. The proof follows immediately from the system of equalities (9). □

Let the relative price vector $\hat{p}={\left\{{\hat{p}}_i\right\}}_{i=1}^n$ be a strictly positive solution of the system of equations

${\hat{p}}_i=\underset{s=1}{\overset{n}{{\displaystyle \sum }}}{\hat{p}}_s{\bar{a}}_{si}+{\hat{\delta }}_i,i=\overline{1,n},$(11)

for some ${\hat{\delta }}_i,i=\overline{1,n}$. Due to the productivity of matrix $\bar{A}$, there exists only one solution of the system of Equation (11).

Proposition 5. Let $p={\left\{p_i\right\}}_{i=1}^m$ be a strictly positive price vector from the subset $R_{+}^m$, and let $\hat{p}={\left\{{\hat{p}}_i\right\}}_{i=1}^n$ be a relative strictly positive solution to the system of Equation (11). Suppose that

${\delta }_i\left(p\left(\hat{p}\right)\right)={\hat{p}}_{f\left(i\right)}{p}_i-\underset{s=1}{\overset{m}{{\displaystyle \sum }}}{\hat{p}}_{f\left(s\right)}{p}_s{a}_{si},i=\overline{1,m},$(12)

are created added values for the price vector $p\left(\hat{p}\right)={\left\{{\hat{p}}_{f\left(i\right)}{p}_i\right\}}_{i=1}^m$. Then the following equalities

$\underset{\left\{i,f\left(i\right)=k\right\}}{{\displaystyle \sum }}{\delta }_i\left(p\left(\hat{p}\right)\right)x_i=X_k{\hat{\delta }}_k,k=\overline{1,n},$(13)

are hold, where $x={\left\{x_i\right\}}_{i=1}^m$ is a strictly positive gross output vector, and $X_k=\underset{\left\{i,f\left(i\right)=k\right\}}{{\displaystyle \sum }}{x}_i{p}_i$.

Proof. It is obvious that

$\begin{array}{c}\underset{\left\{i,f\left(i\right)=k\right\}}{{\displaystyle \sum }}{\delta }_i\left(p\left(\hat{p}\right)\right)x_i=\underset{\left\{i,f\left(i\right)=k\right\}}{{\displaystyle \sum }}{p}_i{\hat{p}}_{f\left(i\right)}{x}_i-\underset{\left\{i,f\left(i\right)=k\right\}}{{\displaystyle \sum }}\underset{s=1}{\overset{m}{{\displaystyle \sum }}}{\hat{p}}_{f\left(s\right)}{p}_s{a}_{si}{x}_i\\ ={\hat{p}}_k{X}_k-\underset{\left\{i,f\left(i\right)=k\right\}}{{\displaystyle \sum }}\underset{l=1}{\overset{n}{{\displaystyle \sum }}}\underset{\left\{s,f\left(s\right)=l\right\}}{{\displaystyle \sum }}{\hat{p}}_l{p}_s{a}_{si}{x}_i\\ ={\hat{p}}_k{X}_k-\underset{l=1}{\overset{n}{{\displaystyle \sum }}}\underset{\left\{i,f\left(i\right)=k\right\}}{{\displaystyle \sum }}\underset{\left\{s,f\left(s\right)=l\right\}}{{\displaystyle \sum }}{\hat{p}}_l{p}_s{a}_{si}{x}_i\\ ={\hat{p}}_k{X}_k-\underset{l=1}{\overset{n}{{\displaystyle \sum }}}\frac{{\hat{p}}_l{\bar{a}}_{lk}}{{\hat{p}}_k}{\hat{p}}_k{X}_k\\ ={\hat{p}}_k{X}_k\left(1-\underset{l=1}{\overset{n}{{\displaystyle \sum }}}\frac{{\hat{p}}_l{\bar{a}}_{lk}}{{\hat{p}}_k}\right)=X_k{\hat{\delta }}_k,k=\overline{1,n}.\end{array}$

□

Further, we assume that the aggregated gross output vector $X={\left\{X_i\right\}}_{i=1}^n$ satisfies the set of equations

$X_k-\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{ki}{X}_i=C_k+E_k-I_k,k=\overline{1,n},$(14)

$X_j-\underset{s=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{sj}{X}_j={\text{Δ}}_j,j=\overline{1,n},$(15)

where

$X_k=\underset{\left\{l,f\left(l\right)=k\right\}}{{\displaystyle \sum }}{x}_l{p}_l,C_k=\underset{\left\{l,f\left(l\right)=k\right\}}{{\displaystyle \sum }}{c}_l{p}_l,E_k=\underset{\left\{l,f\left(l\right)=k\right\}}{{\displaystyle \sum }}{e}_l{p}_l,$

$I_k=\underset{\left\{l,f\left(l\right)=k\right\}}{{\displaystyle \sum }}{i}_l{p}_l,{\text{Δ}}_k=\underset{\left\{l,f\left(l\right)=k\right\}}{{\displaystyle \sum }}{\delta }_l{x}_l,k=\overline{1,n},$

${\bar{a}}_{ki}={\bar{a}}_{ki}\left(p,x\right)=\frac{{\displaystyle \underset{\left\{s,f\left(s\right)=k\right\}}{\sum }{\displaystyle \underset{\left\{r,f\left(r\right)=i\right\}}{\sum }{p}_s{a}_{sr}{x}_r}}}{{\displaystyle \underset{\left\{l,f\left(l\right)=i\right\}}{\sum }{p}_l{x}_l}},k,i=\overline{1,n},$(16)

are the aggregated gross output vector, the aggregated consumption vector, the aggregated export vector, the aggregated import vector and the aggregated matrix of direct costs, correspondingly. We will call the aggregated description of the economic system in the form of (14), (15) by the canonical representation.

A natural question arises whether there is a sustainable development regime for an aggregated description of an economic system based on an price vector $p={\left\{p_i\right\}}_{i=1}^m$ and the aggregation function $f$, mapping the set $M$ onto the set $N$. Below we show that, for the price vector $p={\left\{p_i\right\}}_{i=1}^m$ and the aggregation function $f$, there is always a relative price vector ${\hat{p}}_0={\left\{{\hat{p}}_i^0\right\}}_{i=1}^n$ such that for an aggregated description of the economic system built on the adjusted price vector $p\left({\hat{p}}_0\right)={\left\{p_{f\left(i\right)}^0{p}_i\right\}}_{i=1}^m$ and function $f$, the economic system can be transferred in the mode of sustainable development.

In this section, we will formulate what is meant by sustainable economic development in the case of an aggregate description of the economy. The main principle that we will follow is the principle of correspondence. It means that all the formulas obtained for the taxation system in the aggregate description of the economy should coincide with the formulas obtained in [10] for the taxation system written in value terms. In more detail, the results obtained in [10] for the taxation vector under which there is a sustainable development of the economy are based on the description of the economy in natural terms. The basic principle on which the equilibrium price vector was formed and under which sustainable economic development took place was that there was equality of supply and demand for both resources and manufactured goods. Thus, in this work, our task will be to formulate what the sustainable development of the economy will mean under the aggregated description, and therefore in value indicators. Our first observation is that all the obtained formulas for the vector of taxation in natural indicators in paper [10] can also be presented in value indicators. Therefore, if we consider the relative price vector and accept the hypothesis that the description of the economy in aggregate terms is similar to the description of the economy in natural terms, then the formulation of sustainable development of the economy described in aggregate terms will be the same as in natural terms. We will say that such a formulation of sustainable development of the economy in its aggregate description satisfies the principle of correspondence.

Let the aggregation of the economy into $n$ pure industries take place in relation to the strictly positive price vector $p={\left\{p_i\right\}}_{i=1}^m$, where $m$ is the number of elementary production processes. We suppose that the aggregated matrix $\bar{A}={\Vert {\bar{a}}_{ki}\Vert }_{k,i=1}^n$ is a non negative, productive and indecomposable one. Our assumption relative to the aggregated gross output vector $X={\left\{X_k\right\}}_{k=1}^n$ is that it satisfies the system of equations

$X_k=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{ki}{X}_i+C_k+E_k-I_k,k=\overline{1,n},$(17)

where $C={\left\{C_k\right\}}_{k=1}^n,E={\left\{E_k\right\}}_{k=1}^n,I={\left\{I_k\right\}}_{k=1}^n$ are the aggregated vectors of internal consumption, export and import, correspondingly, relative to a certain strictly positive price vector $p={\left\{p_i\right\}}_{i=1}^m$ from $R_{+}^m$, where

$X_k=\underset{\left\{l,f\left(l\right)=k\right\}}{{\displaystyle \sum }}{x}_l{p}_l,C_k=\underset{\left\{l,f\left(l\right)=k\right\}}{{\displaystyle \sum }}{c}_l{p}_l,E_k=\underset{\left\{l,f\left(l\right)=k\right\}}{{\displaystyle \sum }}{e}_l{p}_l,$

$I_k=\underset{\left\{l,f\left(l\right)=k\right\}}{{\displaystyle \sum }}{i}_l{p}_l,{\text{Δ}}_k=\underset{\left\{l,f\left(l\right)=k\right\}}{{\displaystyle \sum }}{\delta }_l{x}_l,k=\overline{1,n}.$

The aggregated matrix elements ${\bar{a}}_{ki}$ are given by the formula (3).

Let the strictly positive relative price vector ${\hat{p}}_0={\left\{{\hat{p}}_i^0\right\}}_{i=1}^n$ be a solution to the set of equations

$\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{ki}\frac{\left(1-{\pi }_i\right)X_i{\hat{p}}_i^0}{{\displaystyle \underset{s=1}{\overset{n}{\sum }}{\bar{a}}_{si}{\hat{p}}_s^0}}=\left(1-{\pi }_k\right)X_k,k=\overline{1,n}.$(18)

If to consider the aggregation of the economy system relative to the strictly positive corrected price vector $p\left({\hat{p}}_0\right)={\left\{{\hat{p}}_{f\left(i\right)}^0{p}_i\right\}}_{i=1}^m$, where $f$ is an aggregation map, then due to Proposition 2, the matrix elements of the aggregated matrix $\hat{A}={\Vert {\hat{a}}_{ki}\Vert }_{k,i=1}^n$ in this case are given by the formula

${\hat{a}}_{ki}=\frac{{\hat{p}}_k^0{\bar{a}}_{ki}}{{\hat{p}}_i^0},k,i=\overline{1,n}.$(19)

Then the aggregated gross output vector $\hat{X}={\left\{{\hat{X}}_k\right\}}_{k=1}^n$ satisfies the set of equations

${\hat{X}}_k=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\hat{a}}_{ki}{\hat{X}}_i+{\hat{C}}_k+{\hat{E}}_k-{\hat{I}}_k,k=\overline{1,n},$(20)

where $\hat{C}={\left\{{\hat{C}}_k\right\}}_{k=1}^n,\hat{E}={\left\{{\hat{E}}_k\right\}}_{k=1}^n,\hat{I}={\left\{{\hat{I}}_k\right\}}_{k=1}^n$ are the aggregated vectors of internal consumption, export and import, correspondingly relative to the strictly positive price vector $\hat{p}\left({\hat{p}}_0\right)={\left\{{\hat{p}}_{f\left(i\right)}^0{p}_i\right\}}_{i=1}^m$ from $R_{+}^m$, where

${\hat{X}}_k={\hat{p}}_k^0{X}_k,{\hat{C}}_k={\hat{p}}_k^0{C}_k,{\hat{E}}_k={\hat{p}}_k^0{E}_k,{\hat{I}}_k={\hat{p}}_k^0{I}_k,k=\overline{1,n}.$

The set of Equation (18) is written in the form

$\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\hat{a}}_{ki}\frac{\left(1-{\pi }_i\right){\hat{X}}_i}{{\displaystyle \underset{s=1}{\overset{n}{\sum }}{\hat{a}}_{si}}}=\left(1-{\pi }_k\right){\hat{X}}_k,k=\overline{1,n}.$(21)

The last set of equations is similar to the set of equations in value indicators in case of description of an economy in natural indicators [10] . In the following statements, we consider that the aggregation of the economy into $n$ pure industries takes place relative to the price vector $p={\left\{p_i\right\}}_{i=1}^m$, where $m$ is the number of elementary production processes. In all the following statements, we will understand this by default.

Further, under sustainable economic development we understand the growth of the economy’s value indicators in the aggregate description with respect to the corrected price vector $p\left({\hat{p}}_0\right)={\left\{{\hat{p}}_{f\left(i\right)}^0{p}_i\right\}}_{i=1}^m$ from $R_{+}^m$, where the strictly positive relative price vector ${\hat{p}}_0={\left\{{\hat{p}}_i^0\right\}}_{i=1}^n$ solves the set of equation (18) and the vector $p={\left\{p_i\right\}}_{i=1}^m$ is the price vector of aggregation in the economy system.

The principle of correspondence we adopt is formulated in mathematical form in Definition 3 (see also [9] - [12] ).

Definition 3. The taxation system $\pi ={\left\{{\pi }_i\right\}}_{i=1}^m$ can transfer the economic system described in aggregate with respect to the price vector $p={\left\{p_i\right\}}_{i=1}^m$ into the sustainable development mode, if there exists a strictly positive solution of the system of Equation (18) with respect to the vector ${\hat{p}}_0={\left\{{\hat{p}}_i^0\right\}}_{i=1}^n$, which satisfies the conditions

${\hat{p}}_k^0-\underset{s=1}{\overset{n}{{\displaystyle \sum }}}{\hat{p}}_s^0{\bar{a}}_{sk}>0,k=\overline{1,n},$(22)

and such that the economic system described in aggregate with respect to the price vector $p\left({\hat{p}}_0\right)={\left\{{\hat{p}}_{f\left(i\right)}^0{p}_i\right\}}_{i=1}^m$ is in the mode of sustainable development, where $\bar{A}={\Vert {\bar{a}}_{ki}\Vert }_{k,i=1}^n$ is a non negative non-decomposable productive aggregate matrix, $X={\left\{X_k\right\}}_{k=1}^n$ is a strictly positive aggregate output vector with respect to the strictly positive price vector $p={\left\{p_i\right\}}_{i=1}^m$.

The following Definition 4 is important for describing taxation systems that are able to transfer the economic system into the mode of sustainable development.

Definition 4. The taxation system $\pi ={\left\{{\pi }_k\right\}}_{k=1}^n,0<{\pi }_k<1,k=\overline{1,n}$, is able to transfer the economic system into the mode of sustainable development under a strictly positive aggregated gross output vector $X={\left\{X_k\right\}}_{k=1}^n$, if there exists a strictly positive relative equilibrium price vector ${\hat{p}}_0={\left\{{\hat{p}}_k^0\right\}}_{k=1}^n$, solving the system of equations

$\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{ki}\frac{\left(1-{\pi }_i\right)X_i{\hat{p}}_i^0}{{\displaystyle \underset{s=1}{\overset{n}{\sum }}{\bar{a}}_{si}{\hat{p}}_s^0}}=\left(1-{\pi }_k\right)X_k,k=\overline{1,n},$(23)

and such that

${\hat{p}}_k^0-\underset{s=1}{\overset{n}{{\displaystyle \sum }}}{\hat{p}}_s^0{\bar{a}}_{sk}>0,k=\overline{1,n},$(24)

where $\bar{A}={\Vert {\bar{a}}_{ki}\Vert }_{k,i=1}^n$ is a non-negative non-decomposable productive aggregated matrix.

The next Theorem 1 is an auxiliary result which will help to describe taxation systems that are able to transfer the economic systems into the mode of sustainable development.

Theorem 1. Let $Z={\left\{Z_i\right\}}_{i=1}^n$ be a strictly positive vector from $R_{+}^n$ and let $\bar{A}={\Vert {\bar{a}}_{ki}\Vert }_{k,i=1}^n$ be a non-negative non-decomposable aggregated matrix. The set of equations

$\frac{Z_k}{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\bar{a}}_{ki}{Z}_i}}=\frac{{\hat{p}}_k}{{\displaystyle \underset{s=1}{\overset{n}{\sum }}{\bar{a}}_{sk}{\hat{p}}_s}},k=\overline{1,n},$(25)

has a strictly positive solution $\hat{p}={\left\{{\hat{p}}_i\right\}}_{i=1}^n\in R_{+}^n$.

Proof. The proof of Theorem 1 is similar to the proof of Theorem 1 from [10] . □

Consequence 1. The solution of the set of Equation (25) constructed in Theorem 1 is determined uniquely with accuracy up to a positive constant.

The following Theorem 2 and its consequence are important for understanding the possibility to transfer the economic system into the mode of sustainable development by changing the price vector of aggregation.

Theorem 2. Let $p={\left\{p_i\right\}}_{i=1}^m$ be a strictly positive price vector from the subset $R_{+}^m$, and let the relative equilibrium price vector ${\hat{p}}_0={\left\{{\hat{p}}_i^0\right\}}_{i=1}^n$ be a strictly positive solution to the system of equations

$\frac{{\hat{p}}_k^0}{{\displaystyle \underset{s=1}{\overset{n}{\sum }}{\bar{a}}_{sk}{\hat{p}}_s^0}}=\frac{Z_k^0}{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\bar{a}}_{ki}{Z}_i^0}},k=\overline{1,n},$(26)

for a certain strictly positive vector $Z_0={\left\{Z_k^0\right\}}_{k=1}^n$. Then the created aggregated added value

${}_{\widehat{\text{Δ}}}=\underset{\left\{i,f\left(i\right)=k\right\}}{{\displaystyle \sum }}\delta \left(p\left({\hat{p}}_0\right)\right)x_i$(27)

for the vector of prices $p\left({\hat{p}}_0\right)={\left\{{\hat{p}}_{f\left(i\right)}^0{p}_i\right\}}_{i=1}^m$ is given by the formula

${}_{\widehat{\text{Δ}}}={\hat{X}}_k\left(1-\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\bar{a}}_{ki}{Z}_i^0}}{Z_k^0}\right)=X_k\left({\hat{p}}_k^0-\underset{s=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{sk}{\hat{p}}_s^0\right),{\hat{X}}_k={\hat{p}}_k^0{X}_k,k=\overline{1,n}.$(28)

Proof. Since the relative equilibrium price vector ${\hat{p}}_0={\left\{{\hat{p}}_i^0\right\}}_{i=1}^n$ satisfies the set of Equation (26) it also satisfies the set of equations

${\hat{p}}_k^0-\underset{s=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{sk}{\hat{p}}_s^0={\hat{p}}_k^0\left(1-\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\bar{a}}_{ki}{Z}_i^0}}{Z_k^0}\right),k=\overline{1,n}.$(29)

Due to Proposition 5,

$\begin{array}{c}{}_{\widehat{\text{Δ}}}=\underset{\left\{i,f\left(i\right)=k\right\}}{{\displaystyle \sum }}\delta \left(p\left({\hat{p}}_0\right)\right)x_i=X_k{\hat{p}}_k^0\left(1-\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\bar{a}}_{ki}{Z}_i^0}}{Z_k^0}\right)\\ =X_k\left({\hat{p}}_k^0-\underset{s=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{sk}{\hat{p}}_s^0\right),k=\overline{1,n}.\end{array}$(30)

From here, we obtain

${}_{\widehat{\text{Δ}}}={\hat{X}}_k\left(1-\underset{s=1}{\overset{n}{{\displaystyle \sum }}}{\hat{a}}_{sk}\right),k=\overline{1,n},$(31)

where we introduced the denotation ${\hat{a}}_{sk}=\frac{{\hat{p}}_s^0{\bar{a}}_{sk}}{{\hat{p}}_k^0},s,k=\overline{1,n}$. □

Consequence 2. Let the aggregated gross output vector $X={\left\{X_k\right\}}_{k=1}^n$ satisfy to the system of Equation (17), where $C={\left\{C_k\right\}}_{k=1}^n$, $E={\left\{E_k\right\}}_{k=1}^n$, $I={\left\{I_k\right\}}_{k=1}^n$ are the aggregated vectors of internal consumption, export and import, correspondingly, relative to a strictly positive price vector $p={\left\{p_i\right\}}_{i=1}^m$ from the subset $R_{+}^m$. Then the aggregated gross output vector $\hat{X}={\left\{{\hat{X}}_k\right\}}_{k=1}^n$ relative to the price vector $p\left({\hat{p}}_0\right)={\left\{{\hat{p}}_{f\left(i\right)}^0{p}_i\right\}}_{i=1}^m$ satisfies the set of equations

${\hat{X}}_k=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\hat{a}}_{ki}{\hat{X}}_i+{\hat{C}}_k+{\hat{E}}_k-{\hat{I}}_k,k=\overline{1,n},$(32)

${}_{\widehat{\text{Δ}}}={\hat{X}}_k\left(1-\underset{s=1}{\overset{n}{{\displaystyle \sum }}}{\hat{a}}_{sk}\right),k=\overline{1,n}.$(33)

The aggregated matrix elements ${\bar{a}}_{ki}$ are given by the formula (3).

Theorem 3. Let $X={\left\{X_i\right\}}_{i=1}^n$ be a strictly positive gross output vector from $R_{+}^n$ and let $\bar{A}={\Vert {\bar{a}}_{ki}\Vert }_{k,i=1}^n$ be a non-negative non-decomposable productive aggregated matrix. The taxation system $\pi ={\left\{{\pi }_k\right\}}_{k=1}^n,0<{\pi }_k<1,k=\overline{1,n}$, can transfer the economic system into the mode of sustainable development if and only if for it the following representation

${\pi }_i=1-b\frac{{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\bar{a}}_{ij}{Z}_j}}{X_i},0<b<\underset{1\le i\le n}{\text{min}}\frac{X_i}{{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\bar{a}}_{ij}{Z}_j}},i=\overline{1,n},$(34)

is valid for a certain strictly positive vector $Z={\left\{Z_i\right\}}_{i=1}^n\in R_{+}^n$, satisfying the conditions

$\frac{Z_i}{{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\bar{a}}_{ij}{Z}_j}}>1,i=\overline{1,n}.$(35)

Proof. Necessity. Let a taxation system $\pi ={\left\{{\pi }_k\right\}}_{k=1}^n,0<{\pi }_k<1,k=\overline{1,n}$, can transfer the economic system into the mode of sustainable development. Then the equalities (23) are true. From them it follows that

$\left(1-{\pi }_k\right)X_k=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{ki}{Z}_i^0,k=\overline{1,n},$(36)

where

$Z_i^0=\frac{\left(1-{\pi }_i\right)X_i{\hat{p}}_i^0}{{\displaystyle \underset{s=1}{\overset{n}{\sum }}{\bar{a}}_{si}{\hat{p}}_s^0}}>0,i=\overline{1,n},$(37)

${\hat{p}}_0={\left\{{\hat{p}}_i^0\right\}}_{i=1}^n$ is a strictly positive solution to the set of Equation (23). Substituting (36) into (37) we obtain that ${\hat{p}}_0={\left\{{\hat{p}}_i^0\right\}}_{i=1}^n$ solves the set of equations

$\frac{{\hat{p}}_k^0}{{\displaystyle \underset{s=1}{\overset{n}{\sum }}{\bar{a}}_{sk}{\hat{p}}_s^0}}=\frac{Z_k^0}{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\bar{a}}_{ki}{Z}_i^0}},k=\overline{1,n}.$(38)

If to substitute instead of

$\frac{{\hat{p}}_k^0}{{\displaystyle \underset{s=1}{\overset{n}{\sum }}{\bar{a}}_{sk}{\hat{p}}_s^0}},k=\overline{1,n},$

the right side of equality (38) into (23) we get the set of equations relative to the taxation system $\pi ={\left\{{\pi }_k\right\}}_{k=1}^n,0<{\pi }_k<1,k=\overline{1,n}$,

$\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{ki}\frac{\left(1-{\pi }_i\right)X_i{Z}_i^0}{{\displaystyle \underset{s=1}{\overset{n}{\sum }}{\bar{a}}_{ij}{Z}_j^0}}=\left(1-{\pi }_k\right)X_k,k=\overline{1,n}.$(39)

The solution of the set of Equation (39) is

${\pi }_k=1-b\frac{{\displaystyle \underset{i=1}{\overset{n}{\sum }}{\bar{a}}_{ki}{Z}_i^0}}{X_k},0<b<\underset{1\le k\le n}{\text{min}}\frac{X_k}{{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\bar{a}}_{kj}{Z}_j^0}},k=\overline{1,n}.$(40)

Due to the fact that the relative equilibrium price vector ${\hat{p}}_0={\left\{{\hat{p}}_i^0\right\}}_{i=1}^n$ also satisfies the set of equations (38) it follows that the inequalities (35) are true, since the relative equilibrium price vector ${\hat{p}}_0={\left\{{\hat{p}}_i^0\right\}}_{i=1}^n$ satisfies the inequalities (24).

Sufficiency. Suppose that the taxation system is given by the formula (34) for which the inequalities (35) are true. Let us prove the existence of strictly positive solution ${\hat{p}}_0={\left\{{\hat{p}}_i^0\right\}}_{i=1}^n$ of the set of equations (23) satisfying the inequalities (24). Substituting (34) into (23), we obtain the set of equations relative to the equilibrium price vector ${\hat{p}}_0={\left\{{\hat{p}}_i^0\right\}}_{i=1}^n$

$\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{ki}\frac{b\underset{j=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{ij}{Z}_j{\hat{p}}_i^0}{\underset{s=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{si}{\hat{p}}_s^0}=b\underset{j=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{kj}{Z}_j,k=\overline{1,n}.$(41)

Let us choose the vector ${\hat{p}}_0={\left\{{\hat{p}}_i^0\right\}}_{i=1}^n$ such that it would satisfy the system of equations

$\frac{{\hat{p}}_i^0}{\underset{s=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{si}{\hat{p}}_s^0}=\frac{Z_i}{\underset{j=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{ij}{Z}_j},i=\overline{1,n}.$(42)

Then, it will satisfy the set of Equation (23). But the strictly positive solution to the set of Equation (42) always exists, due to Theorem 1. Owing to the inequalities (35), the inequalities (24) are true.

Note 1. If the matrix $\bar{A}$, in addition, is non degenerate, then under taxation system given by the formula (34) satisfying the conditions (35), the relative equilibrium price vector must satisfy the set of equations (42).

The Note 1 means that if to introduce the taxation system, given by the formulas (34), (35), then in the economic system the aggregation will be described by the vector $p\left({\hat{p}}_0\right)={\left\{p_{f\left(i\right)}^0{p}_i\right\}}_{i=1}^m$.

Note 2. There always exist strictly positive vector $Z={\left\{Z_k\right\}}_{k=1}^n$ satisfying the conditions (35) if the conditions of Theorem 3 are true.

Proof. Let choose the vector $Z={\left\{Z_k\right\}}_{k=1}^n$ solving the set of equations

$Z_k=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{ki}{Z}_k+F_k,k=\overline{1,n},$

where $F_k>0,k=\overline{1,n}$. Since the matrix $\bar{A}={\Vert {\bar{a}}_{ki}\Vert }_{k,i=1}^n$ is productive, the solution of the set of equations there always exists, which satisfies the conditions (35). □

It follows from Note 2 and Theorem 3 that there always exists a tax system such that for any strictly positive price vector $p={\left\{p_k\right\}}_{k=1}^m$ of aggregation, the economic system described in aggregate can be transferred into a sustainable development regime.

Next Theorem 4 states that there are no taxation systems other than those described above that can transfer an economic system into the mode of sustainable development.

Theorem 4. Let $X={\left\{X_i\right\}}_{i=1}^n$ be a strictly positive gross output vector from $R_{+}^n$ and let $\bar{A}={\Vert {\bar{a}}_{ki}\Vert }_{k,i=1}^n$ be a non-negative non-decomposable productive matrix. An economic system can be transferred into the mode of sustainable development if and only if for the taxation system $\pi ={\left\{{\pi }_k\right\}}_{k=1}^n,0<{\pi }_k<1,k=\overline{1,n}$, the representation (34) is true, which satisfies the conditions (35).

Proof. The proof of the Theorem 4 is analogous to the proof of Theorem 3 from [10] . □

The following Theorem gives us a new conditions under which an economic system can be transferred into the mode of sustainable development.

Theorem 5. Let $X={\left\{X_i\right\}}_{i=1}^n$ be a strictly positive gross output vector from $R_{+}^n$, such that the vector $\left(1-\pi \right)X={\left\{\left(1-{\pi }_k\right)X_k\right\}}_{k=1}^n$ belongs to the interior of the cone created by the column vectors of the matrix $\bar{A}{\left(E-\bar{A}\right)}^{-1}$, where $\bar{A}={\Vert {\bar{a}}_{ki}\Vert }_{k,i=1}^n$ is a non-negative non-decomposable productive matrix. For the taxation system, given by the formula (34), satisfying the conditions (35), the economic system, described by “input-output” production model, can be transferred into the mode of sustainable development.

Proof. From the taxation system, given by the formula (34), we have

$\left(1-{\pi }_k\right)X_k=b\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{ki}{Z}_i^0=\underset{i=1}{\overset{n}{{\displaystyle \sum }}}{\bar{a}}_{ki}{Z}_i,k=\overline{1,n},$(43)

for a certain strictly positive vector $Z_0={\left\{Z_i^0\right\}}_{i=1}^n$, where we put $Z={\left\{Z_i\right\}}_{i=1}^n=b{\left\{Z_i^0\right\}}_{i=1}^n$. Thanks to Theorem 4, the relative equilibrium price vector ${\hat{p}}_0={\left\{{\hat{p}}_i^0\right\}}_{i=1}^n$ solves the set of equations

$\frac{{\hat{p}}_i^0}{{\displaystyle \underset{s=1}{\overset{n}{\sum }}{\bar{a}}_{si}{\hat{p}}_s^0}}=\frac{Z_i}{{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\bar{a}}_{ij}{Z}_j}},i=\overline{1,n}.$(44)

Due to the conditions of Theorem 5, for the vector $\left(1-\pi \right)X$ the representation

$\left(1-\pi \right)X=\bar{A}{\left(E-\bar{A}\right)}^{-1}\alpha ,\alpha ={\left\{{\alpha }_k\right\}}_{k=1}^n,{\alpha }_k>0,k=\overline{1,n},$(45)

is true. The representation (45) for the vector $Z={\left\{Z_i\right\}}_{i=1}^n$ gives us the formula $Z={\left(E-\bar{A}\right)}^{-1}\alpha$ for a certain strictly positive vector $\alpha ={\left\{{\alpha }_i\right\}}_{i=1}^n$. From this it follows that for the vector $Z={\left\{Z_i\right\}}_{i=1}^n$ the inequalities

$\frac{{\hat{p}}_i^0}{{\displaystyle \underset{s=1}{\overset{n}{\sum }}{\bar{a}}_{si}{\hat{p}}_s^0}}=\frac{Z_i}{{\displaystyle \underset{j=1}{\overset{n}{\sum }}{\bar{a}}_{ij}{Z}_j}}>1,i=\overline{1,n},$(46)

are true. This proves Theorem 5. □

Theorem 6. Let $\bar{A}={\Vert {\bar{a}}_{ki}\Vert }_{k,i=1}^n$ be a non-negative non-decomposable productive matrix and let $X={\left\{X_i\right\}}_{i=1}^n$ be a strictly positive gross output vector solving the system of Equation (17). Suppose that the economic system can be transferred into the mode of sustainable development by the taxation system (34), satisfying the conditions (35). Then the set of pure branches $N=\left\{1,2,\cdots ,n\right\}$ is divided into two disjoint sets $I$ and $J$ such that $I\cup J=N$ and the inequalities

${}_{\widehat{\text{Δ}}}\le {\hat{C}}_k+{\hat{E}}_k-{\hat{I}}_k,k\in I,{}_{\widehat{\text{Δ}}}>{\hat{C}}_k+{\hat{E}}_k-{\hat{I}}_k,k\in J,$(47)

are valid, where

${\hat{X}}_k=X_k{\hat{p}}_k^0,{\hat{C}}_k=C_k{\hat{p}}_k^0,{\hat{E}}_k=E_k{\hat{p}}_k^0,{\hat{I}}_k=I_k{\hat{p}}_k^0,$