Theoretical models, which are simplified representations of real systems and processes, are used for their description and analysis. These models help to better understand, analyze, control, or predict the development of the phenomena being studied.
The course is focused on introducing selected mathematical concepts, problems, and methods used for modelling real-world processes in economics.

Upon successful completion of this course, students will gain an overview of mathematical methods, which will serve to better understanding of economics, and will be able to use the acquired mathematical knowledge in solving selected economic problems.

1. Symmetric matrices and quadratic forms. Positive and negative definiteness of a quadratic form.

2. Differentials of multivariate functions.

3. Optimization problems in economy and the mathematical problem on constrained extrema. Lagrange's method of solution and its economic interpretation.

4. Basics of convex analysis. Convex functions and their properties. Cobb-Douglas production functions.

5.  Nonlinear programming. Lagrange's method. Kuhn-Tucker theorem.

6. Convex programming. Quadratic programming. Applications in economy.

7. Nonlinear programming in economy. 

8. Basics of game theory. Overview of various types of problems (antagonistic games, games with two and more players, games with constant and nonconstant sum, matrix games). Prisoner's dilemma.

9. Matrix games with constant sum. Pure strategies, saddle points, minimax.

10. Matrix games with constant sum. Mixed strategies.

11. Models of oligopoly. The Nash equilibrium.

12. Basics of  differential and difference equations, their application in modelling of continuous and discrete dynamic processes in economy.

13. A review of the main mathematical notions, tools and models.

1. Matrices and determinants. Functions of two variables.

2. Symmetric matrices and quadratic forms. Verification of the definiteness and indefiniteness of a quadratic form.

3. The meaning of the differential of a function of two variables. Second order differential and its computation.

4. Lagrange's method for the problem on constrained extrema.

5. Convex functions. Verification of convexity using differential calculus.

6. Lagrange's method in nonlinear programming.

7. Convex programming.

8. Nonlinear programming in economy.

9. Basics of game theory. Overview of various types of problems, examples.

10. Matrix games with constant sum. Pure strategies, saddle points, minimax.

11. Matrix games with constant sum. Mixed strategies.

12. Models of oligopoly. The Nash equilibrium.

13. Basics of differential and difference equations, examples.

Learning Outcomes:

Professional Knowledge
The student has an overview of mathematical concepts, problems, and methods commonly used in economic modeling, which enables a deeper understanding of economic phenomena.

Professional Competence
The student is able to select and apply appropriate mathematical methods to solve models of real-world processes and to interpret the results within the context of the given application.

Professional Skills
The student is capable of solving basic nonlinear programming problems that arise in modeling real-world processes, makes use of computational tools, and is able to interpret the results effectively.