5 semester


Name of the discipline



Course of Study

3, mathematics (economic activity)


Semester of training



Amount of credits



Full name of lecturer

Zabreiko Petr Petrovich


Objectives of studying the discipline

the formation of the fundamentals of mathematical thinking among students, mastering the methods of mathematical modeling of economic phenomena and processes, as well as the ability to interpret the results of research of mathematical models from the economic point of view.

As a result of studying the discipline, the trainee must

be able to:

– to compile mathematical models on economic data technological matrices;

– to check the productivity of technological matrices, to evaluate the effectiveness of the relevant models; determine positive and non-negative eigenvectors of technological matrices;

– calculate or evaluate the main characteristics of technological matrices;



Mathematical analysis


Probability theory

Differential equations,

The theory of functions of a complex variable,

Functional Analysis


Contents of the discipline

Theme 1. Economics and mathematics

Mathematics and Economics. The space of goods; basic structures of finite-dimensional space from the economic point of view. Coordinates in “economic” spaces; “permissible” changes of variables. Dual space from an economic point of view. Real and complex spaces. Time in the economy. Discrete and continuous time. General scheme of economic relations. Producers and consumers. Markets. Static and dynamic models in the economy.

Theme 2. The open model of Leontief’s productivity

Open model of Leontief. The Leontief equation. Production matrix. The economic meaning of the elements, rows and columns of the production matrix. Nonreducible and decomposable matrices. Economic sense. Productive open models of Leontief. Productivity conditions. Spectral radius, its basic properties and computation. The Neumann series.

Volumes of production and consumption. Production costs. Direct and indirect, total costs. The main theorems on the cost of production in the case of an indecomposable technological matrix and in the general case. Topic 3. Non-negative matrices and spectral radius Positive and non-negative matrices, graph of a nonnegative matrix. The spectral radius of the matrix, various formulas for the spectral radius: the Neumann series, the Gel’fand formula, the spectral radius coupling with equivalent norms

Theme 4. The Perron-Frobenius themes

Spectral properties of positive matrices. The existence of a positive eigenvector and the positivity of the spectral radius. Minimax formulas for the spectral radius of a positive matrix. Non-decomposable and decomposable matrices and their spectral properties. Boolean matrices and their applications to the theory of nonnegative matrices.


Recommended literature

1. Allen R. Mathematical economics. – Moscow: Foreign Literature Publishing House, 1963. – 600 p.

2. Ashmanov S.A. Mathematical models and methods in economics. – Moscow: Publishing house of Moscow University, 1980. – 200 p.

3. Ashmanov S.A. Introduction to mathematical economics. – Moscow: Science, the main edition of physics and mathematics, 1984. – 294 p.

4. Voevodin VV, Kuznetsov Yu.A. Matrices and calculations

(Reference and Mathematical Library).} – Moscow: Nauka, Home

edition of physical and mathematical literature, 1984. – 320 p.

5. Gantmakher FR Theory of matrices. – Moscow: Science, Fizmatgiz, 1966. – 548 p.

6. Dorfman R., Samuelson P.A., Solow R.M. Linear Programming and Economic Analysis. – New York Toronto London: McGraw-Hill Book Company, Inc., 1958. – 525 p.

7. PP Zabreiko Mathematical Basics of Economics. – Lectures, 2002, 1-54.

8. Zabreiko PP, Shevelevich K.V. Theorems of Hicks and Le Chatelier-Samuelson for decomposable nonnegative matrices. – Reports of the National Academy of Sciences of Belarus, (2002), No 3, p. 30-34.

9. Intriligator M. Mathematical methods of optimization and economic theory. – Moscow: Progress, 1975. – 608 p.

10. Karlin S. Mathematical methods in the theory of games, programming and economics. – The World, 1964. – 839 p.

11. Kuhn HU, Tucker A.U. Linear inequalities and related

questions. – M .: Publishing house of foreign literature, 1959. – 470 p.

12. Lancaster, K. Mathematical Economics. – Moscow: Soviet Radio, 1972. – 464 p.

13. Sydsetter K., A., Burke P. Handbook on

mathematics for economists. St. Petersburg: Economic School. St. Petersburg State University of Economics and Finance. Higher School of Economics, 2000. 1-230

14. Horn R., Johnson C. Matrix analysis. – Moscow: Mir,

1989, – 656 p.

15. Arrow K.D., Hurwitz L., Udzawa H. Studies on

linearity and nonlinear programming. – Moscow: The Publishing House of Foreign Literature, 1962. – 333 p.


Teaching Methods

interactive methods of teaching (working in small groups (team), problem training) are organized taking into account the inclusion in the learning process of all students of the group. Joint activity means that each student makes his own individual contribution, in the course of the work there is an exchange of knowledge, ideas, methods of activity. Organized individual, steam and group work. Interactive methods are based on the principles of interaction, activity of trainees, reliance on group experience, mandatory feedback


Language of education



Conditions (requirements), current control

– colloquium;


Form of certification