Can we, people of the 21st century, imagine our world without mathematics? Of course, we can’t do this because no discipline (science) can do without accurate calculations and analysis and at the same time not using mathematics. Many people think that mathematics is the queen of sciences and it is the language of all accurate sciences. Physics, chemistry, biology, etc., rely on it. Economics is no exception. You can find widespread mathematics in economics from modeling processes in the economy to the mathematical analysis of the model you got. The main purpose of this article is to show the potential of applied mathematics in economics. In other words we are going to consider mathematical economics.
First of all, we need to give an exact definition of this concept. According to the dictionary ‘mathematical economics is a model of economics that utilizes math principles and methods to create economic theories and to investigate economic quandaries. Mathematics permits economists to conduct quantifiable tests and create models to predict future economic activity.’ Mathematical economics includes econometric, mathematical optimization, game theory, linear and nonlinear programming, function analysis, and go etc. Also, it is worth mentioning financial mathematics.
What should we start with? Primarily, we need to build a model of an investigated object and to describe how this object behaves. But we can’t do it accurately because we can rely only on past data in a similar situation. We need to use simplified assumptions to make the model. It is one of the basic principles of economics. After we made the model we can analyze it and get a new useful information. For example, we have producers (firms) and consumers and we want to describe how they make decisions in the world of scarcity. For that, we need to build a model of their behavior by using data from similar situations in the past. But this is not enough because we need more data. So we suggest that producers always maximize profit and consumers maximize the utility of goods they buy. As practice shows, in most cases, this suggestion is justified.
Now let's give a brief description of econometrics as a part of mathematical economics. It is the application of statistical methods of economic data. In other words, in order to carry out any analysis in the economy, data are required. A basic tool for econometrics is the multiple linear regression model. A striking example of the application of econometrics is the method of least squares. The essence of this method is that it lets us know the values of some variables from the previous similar situation and we need some parameters of the function (for example, the parameters of the demand function for the goods). We minimize the sum of squared deviations of the function we want to construct from the variables sought. Thus we get the most approximate to the reality the parameters of the function.
Financial mathematics deals with interest rates, financial flows, company activity analysis, etc. This is a very important subject in economics. Mainly financial mathematics shows the difference between present value and future value of the investigated object (it can be finance flow or something else). This difference is very significant because time plays an important role in the economy (time is money) and money today is not equal to money tomorrow. For example, we need to find a break-even point. There are two approaches to that. Using the first of them, let's call it accounting, we take into account only depreciation. In the second approach, we take into account possible investments and subtract them from the total amount of the flow. It is the second method that gives a more accurate assessment of the situation. For example, Lockheed company appealed in 1971 to the U.S. Congress about the loss of production of military aircraft TriStar L-1011. The appeal was argued that the commercial attractiveness of the production was determined taking into account the break-even point of production of about 200 aircraft. However, this figure did not take into account the previous investment of $ 1 billion. Taking into account these imputed costs, the break-even point is increased to 500 aircraft).
Mathematical Optimization is a branch of applied mathematics that is useful in many different fields. In the framework of this article, it is simply impossible to fully describe this concept. Let we need to maximize our profit and we have the profit function of the goods that the company produces. We can get extremum points of our function using methods of mathematical optimization. Then we find the point of the global maximum of the function and this point is the solution to the problem. Using the methods of mathematical optimization important discoveries in Economics were obtained. For example, let us have utility function U(X, Y) and we know the price values X, Y of these products. Construct the Lagrange function:
Find the partial derivatives of this function:
Divide equation number 2 by equation number 1 and transform this system of equations:
This is Gossen’s second law.
It is also worth separating linear and nonlinear optimization (an example of linear optimization was given above). Also, we can see this fact if we will look at the function U(X, Y) graph.
Game theory is a theoretical framework for conceiving social situations among competing players. In some respects, game theory is the science of strategy, or at least the optimal decision-making of independent and competing actors in a strategic setting. The key pioneers of game theory were mathematicians John von Neumann and John Nash, as well as economist Oskar Morgenstern. The simplest example describing what game theory explores is the prisoner's dilemma. Two men may have committed a robbery, but the police have no evidence against them. The police are going to question each of them separately. Each of them has two options. The first is to be silent. The second is to give his accomplice. There are 4 possible outcomes. The first outcome is that both of them remained silent and, since the police have no evidence on them, but they are the main suspects, they are imprisoned for 1 year. The second and third outcomes are almost identical and consist in the fact that one of the prisoners betrays the other and one of them is given 10 years, and the other is released. The fourth outcome is that they both give each other away and they are given 5 years in prison. Despite the fact that the first outcome is the most profitable, within the framework of game theory it is proved that the most frequent outcome is the fourth outcome. Game theory is probably one of the youngest and most unexplored areas of mathematics. Today specialists in this area are appreciated by many universities and corporations.
Concluding this article it is worth saying that there is a huge prospect for the application of mathematics in Economics. Only recently, a lot has been discovered. Each year, the Nobel prize in Economics is awarded for the most important discovery. In 2018, the Nobel prize in Economics was awarded to American economists William Nordhaus and Paul Romer. They are recognized for achievements in long-term macroeconomic analysis. All these discoveries bring us one step closer to the truth, opening up new opportunities for making the most correct decisions in the economic sphere.
5. Четыркин Е.М. Финансовая математика: Учебник. — 5-е изд., испр. — M: Дело, 2005. - 400 с. ISBN 5-7749-0193