History of mathematical modelling
The beginnings of mathematical modelling of economic phenomena can be found in the very early stages of classical political economy development. English scholar William Petty (1623-1687) emphasised measurement of values and schematic modelling of relationships between them in his study of national economy. French economist Francois Quesnay (1694-1774) proposed a tabular representation of the national economy which is considered as the first macroeconomic model.
Leon Walras (1834-1910) used mathematical apparatus as absolutely essential part of his considerations about economic theory of marginal utility and by derivation of the general theory of economic equilibrium. His pupil Vilfredo Pareto (1848-1923) was a successor in his professorship at the University of Lausanne, whose name is today primarily associated with Pareto’s optimality.
The development of mathematics in economics continued to progress in many ways after the First World War. An important milestone in this development was the year 1931 when the Econometric Society was founded. After that the society began to periodically publish journal Econometrica. This journal started to shape the science of econometrics. The mathematical description and statistical verification of economic relations and the concept of introduction of mathematical methods to economics are considered as the main goals of econometrics.
In 1939, Leontief published his work The Structure of American Economy, which became the basis for structural analysis. In the same year, L.V. Kantorovich formulated the principles of linear programming. In 1944 J. von Neumann and O. Morgenstern published the book Theory of Games and Economic Behavior.
After World War II the development of linear programming is regarded to be the credit of G.B. Dantzig, the author of the simplex method. In the fifties, operations research, including mathematical programming, network analysis, queueing theory, inventory theory, simulation models, and game theory, has become an established discipline and in the sixties and seventies multi-criteria optimization has emerged.
- Mathematical economics is a set of quantitative methods of modern economic theory that capture mathematical problems of economic balance.
- Econometrics is a discipline that uses economic, mathematical, and statistical methods for searching, measuring, and verification in inter-relationships between economic variables.
- Structural analysis is a method for analysis of links between elements of a particular economic system and their linkage between the system and surroundings.
- Linear programming deals with problems of formulation and solution of mathematical models that are linear, static and deterministic, and that represent the observed dependence by using linear functions.
- Queuing theory quantitatively examines processes in which particular units might accumulate due to limited capacity of service.
- Inventory theory is used to optimize the management for the processes of creating, completing, maintaining and drawing stocks.
- Game theory deals with the classification and modeling of conflict situations and searching for optimal strategic processes in different types of situations. It is closely related to the oligopoly theory.
- Dynamic programming is a special technique for searching of extremes, which converts multi-object problems into maller sub-problems.
- Network analysis is a class of models and methods based on the graphic expression of complex projects and deals with analysis of these projects in terms of time, cost or resources.
- Integer programming introduces the constraints for integer values to the linear programming. The variable can take only the values 0 or 1 for bivalent values.
- Multi-objective optimization seeks compromise solution in decision-making situations that are constrained by several evaluation criteria.
- Simulation models deal with complex dynamic structure models of probabilistic systems that are analytically intractable. Results are obtained by computer experimentation with the model.