Structural management models of technological innovations

In order to define the parameters of structural innovations of the technological core of the economic system, a formalized criterion of the effectiveness of these innovations has been proposed, a model of the technological core has been developed, as well as mathematical methods of its analysis. The developed model uses the cross-sectoral balance sheet of the national accounts of the economy. The analysis of the model consists in calculating the preferred structure of the technological core and calculating plans for its phased development.

The effectiveness of structural innovation can be measured by the relative value added of these innovations per unit of direct cost. This makes it possible to use as a measure of the efficiency of the technological core of the economy; its productivity -the share of value added per unit of direct cost. Using a closed-loop inputoutput model of the Leontiev type [2], the productivity of the technological core of the economic system shall be determined by analogy with the efficiency factor in the technique as ʌ = Y/Z, where Z is the intermediate input, Y is the value added (GDP), V -gross output. Denote material intensity a = Z/V, gross output V = Y + Z. Then: ʌ = (V-Z)/Z = 1/a -1. Since different states of the economic model have different levels of productivity, the choice of the most productive model can be considered.
The productivity potential of the technological core of an economic system is defined for the economic model as: The multipurpose production model [3] defines the output multiplier (the indicator of the productivity of the economic system) as a function of the structural proportions of output and prices of output and service industries. The maximization of this output-to-input ratio determines a balanced output-to-price structure that corresponds to a reproduction regime, where growth shares for all products and services are the same.
It is assumed that Zij is the direct input of industry j on output of type i products or services, Vj is the output of products and services of type j. On the basis of these data aij -the coefficients of unit input are calculated: The input-output model can be represented by the following ratio: from where the stationary reproduction process can be represented by a ratio: γ is a multiplier of the output of the industry i, which denotes the excess of industry output over inputs.
Denote γ -the minimum by industry multipliers: The formulation of the optimization problem for the output structure vector V with the minimum output multiplier maximum criterion γ is as follows: γ Vi max with technological restrictions on production: This task allows defining the equilibrium system of releases.
In target setting, when final consumption, exports and accumulation are formally part of an activity, the stationary regime allows a change in the share of output in those areas. Otherwise, the values 1 − i γ are the shares of output of industries used for final consumption, exports and savings. To maintain these shares equal to the reference values the optimization problem for the release structure will take the form of: , maxγ . Statement 1. If the matrix A is square, the solution to the problem of finding an equilibrium output system is its eigenvector.
The release growth condition is: Let A be a non-negative matrix whose maximum modulo eigenvalue is Ȝ, |Ȝ|<1. The eigenvector x of this matrix satisfies the equation: The eigenvector x with a modular maximum eigenvalue is searched by an iterative procedure: This result shows that the inertial process of reproduction of the technological nucleus has a stable equilibrium with a fixed point equal to the eigenvector x .

Method of evaluation new technologies by productivity index
is the final consumption of the service , the unit cost factors are calculated by the formula )) The new technology with number ν adds to the cross-sectoral balance sheet: • a new line of its costs for other services and industries; • a new column of this technology in other industries and final consumption. Then the performance criteria of the new technology will be the fulfillment of inequalities:

Conversion of Output Indices
When the output volumes change, the unit cost estimates ij a change too. In order to capture the results of the change the volumes i V in the previous cycle, the direct cost factors are converted: The transformation operations of the process matrix in current prices to the matrix in relative prices and vice versa are considered.
We denote D -diagonal matrix with diagonal , then a transformed matrix with coefficients has the same eigenvalue as matrix A, and the eigenvector is equal to the original one up to the multiplier D .
Let us call this process matrix transformation D a deformation transformation. To move to the prices on a relative scale (in proportion to the prices), the eigenvector of outputs 0 V on an absolute scale is used by deforming the process matrix: In this case, the eigenvector of the process matrix becomes unit after deformation. The eigenvalue will be maintained and the eigenvector * v of the matrix 1 A in relative scale can be converted to absolute scale by conversion: In the calculation of values i V , the values are interpreted as volume indices of the services to be performed, and the limitations on them in the assumption of non-discharge are in the form of . ,...,

Indicative projection of structural innovation
It is not possible to instantly implement a change in the structure of the releases, making them equilibrium. In order to determine the most rational plan for the development of the industry, the following local objective may be used: , maxγ Vi technological restriction on production: . ,..., 1 ), By repeating the procedures for finding an optimal solution and converting the matrix of direct costs from tact to bar, we get an indicative multi-stage plan-forecast of the joint development of branches of the technological core of the economy. The indicative plan calculation procedure uses absolute and relative output values. If 1 V -current price vector, then in the first step the process matrix deformation is used to shift to relative output 1 v : where I is the unit vector. Then the following statements can be used to calculate the indicative projection plan for joint industry development.
Statement 4. The sequence i V for a finite number of bars goes to the proper vector of process matrix A, and the evaluation i γ goes to the proper number of that matrix.
Note. When converting process matrix A with matrix deformation diag(v*) D = , the solution to the planning problem becomes trivial: . That is, when achieving technological equilibrium, the structure of the output does not change further.
Statement 5. If there is a solution to local problems at 1 ≥ i , the quantity of indicative releases in absolute units of the following type can be obtained: Statement 6. If for all steps the coefficient is constant, the relative increase in output from a certain clock becomes equal. This property is similar to the central property of optimization models of economic dynamics [4].
If x -the resulting eigenvector, then for x y≠ the value of the estimate y / Ay = a can be reduced, this corresponds to an increase in the productivity estimate π .
In order to find an output vector that maximizes the productivity of the technological kernel, we will apply a gradient descent procedure. Let A be a positive matrix with non-negative components. Express material content a = Z/V in terms of the technology nucleus model: where the vector norm is calculated as the average: Productivity assessment can be presented as: Statement 7. We denote I -unit vector, E -diagonal unit matrix, h -step value, Minimization procedure: The gradient descent method is: where k is the iteration number.
The choice of the step h determines the speed of convergence of the process: the more it is the greater the speed of convergence. However, infinite enlargement does not accelerate convergence.
The process of optimizing the output structure produces a sequence of incremental assessments of the productivity of the technological core, accompanied by a sequence of increments in the output of certain industries. At the same time, the increase in output from other industries does not lead to an increase in productivity assessment. The rate of convergence depends on the size of the step h : first, when the step is increased, the speed is increased.

Expenditure accounting for non-critical, but significant industries
We consider a list Ξ of activities of significance for the state, but with small values of equilibrium indices of output and not belonging to the whole «bottleneck» (non-critical, relevant branches: science, culture, social sphere, safety, ecology): Ξ ∈ i . An additional constraint to the indicative planning task is: Fig. 1 shows the dependence of the output of the three branches (computing, communications, education) on the cost of research with the optimal output distribution.

Fig. 1. Impact of research expenditure on technological equilibrium
This correlation shows that the multiplication of research costs across a wide range has little impact on the equilibrium of inputs and outputs in other industries. The swings at the 20 and 35 levels are explained by the reserve of the industry's imperfections and by the fact that only a 20-and 35-fold increase in the cost of science, respectively, could change the structure of other industries output and require a further increase in the gross output of the economy as a whole.

Results. Calculation of the projection plan
The following is a graph of the evolution of the productivity factor of the technological core in the balancing of output at successive stages of indicative planning, where the lower bound of the ratios of output was 1 and the upper limit of the variation in the ratios of output was = Δθ 1.5 to the stage. The following is a graph of the indicative development of the proportions of output of some industries in solving the problem at successive stages of indicative planning for data on the Russian economy.

Conclusion
Despite its high potential for development, the modern economy of the Russian Federation is facing crisis phenomena. At the macro level, these include low GDP growth, the economy's critical dependence on oil and gas exports, exchange-rate volatility and undervaluation, a small share of manufacturing, dependence on external sanctions, and poor governance. These factors contribute to the under-realization of the potential of the technological core of the economy [5]. The results show the possibilities for improving the economy's efficiency, based on a gradual change in the structure of its technological core. Based on Rosstat's data, possible growth in economic productivity could be more than doubled. The practical realization of this possibility should be linked to the formulation of strategic plans for the development of the economy. And, in addition to the selection of priority areas for the development of the technological nucleus, it requires the application of adequate methods for forecasting multisectoral dynamics. This takes into account all major aspects of economic activity: stock formation, accumulation, final consumption of the State and households, export-import flows [6,7]. Planning at a new level also involves appropriate institutional arrangements [8].