Fundamental properties of the main matrix forms in the systems of equations of intersectoral balance
Journal Title: Проблеми загальної енергетики - Year 2017, Vol 0, Issue 2
Abstract
We have investigated the fundamental properties of the matrices of three systems of algebraic equations describing the key problems of intersectoral balance: determination of output by the data of final demand, determination of output by the data of added value, and establishing the interrelation between equilibrium prices and output. All these problems have been solved with using the same method proposed here and called the method of extrapolation to zero determinant. We have shown that the system of equations recommended by numerous authors for establishing the interrelation between equilibrium prices and output volumes in output units is homogeneous. We have proved that, in this matrix, there exist at least n positive minors with a dimension r = n - 1 , where n is the dimension of matrix. This important feature envisions the fact that, in the continual set of solutions of the corresponding system, it is impossible to find even if a single vector that would correspond to the meaning content of tables “input-output”. Therefore, this system cannot be used not only for determining equilibrium prices and output in output units, but also for the solution of other problems of intersectoral balance. We have proved that the matrix of the system of equations for finding output by the data of final demand has a rank r = n, and its determinant is always positive for any dimension of this matrix and non-negativeness of the elements of final demand, and the last condition not always is necessary. We have established that the system of equations for finding output by the data of added value has a matrix whose rank is r = n, and its determinant is positive for all values of variables satisfying the condition of input balance. The property of positiveness of the determinant of this matrix was already proved by R. Bellman, and this proof was based on the Gram determinants. It is more compact than the method of extrapolation to zero determinant, but the latter can be applied to all problems considered here and, which is quite important, enables one to obtain not only the sign, but also the values of determinants under study. We should especially emphasize the unique property of matrices of the systems of equations for determining output by the data of final demand and added value. The determinants of these matrices depend on the values of right-hand sides of the corresponding systems, namely, on the values of elements of the vectors of final demand and added value, respectively. The initial cause of such property is the fact that these systems were constructed on the basis of output and input balances.
Authors and Affiliations
Myhajlo Kulyk
Keywords
Related Articles
- EP ID EP456816
- DOI 10.15407/pge2017.02.014
- Views 74
- Downloads 0
Forecasting the efficiency of work of coal mines in Ukraine
Coal is at present and will remain in visible future the main fuel resource for thermal and electric generation. However, the complex mining and geological conditions of coal occurrence in Ukraine, the deterioration of m...
Assessing the possibilities for improving energy efficiency of TPPs based on flue gas heat exchangers and using heat pump installations
Possible volumes of exhaust flue gas heat recovery by using heat pump installations at Ukrainian TPPs have been determined. The heat recovery system diagram provides for a waste heat exchanger installed at the flue gas p...
The influence of the work in the maneuvering regimes of power units of coal-firing generating station on the technological and economic characteristics
There is a necessity to provide the peak of load energy in Ukraine. Because of absence of gas turbines and insufficient capacity of hydroelectric power plants, the large energy blocks, which were designed for working in...
Advanced technical approach to identify appropriate energy saving potential in energy-intensive economic activities
It was improved the technique for determining the economically appropriate energy saving potential due to changes in the registering the statistical reporting (introduction of KVED-2010 to replace KVED-2005) and in law (...
Analysis of the demand of Ukrainian thermal power plants for steam coal with regard for requirements to fuel quality
The thermal power industry of Ukraine is based on 14 large power plants working on coal. The boilers of seven plants were designed to burn coal of the gas group and the other seven to work on anthracite fuel. The use...