Thesis "Application of Mathematics in Economy"

Reflections on the application of mathematical methods in economic research

How to understand the application of mathematical methods in economic research has always been controversial in academic circles. Since 1969 awarded the first Nobel Prize in Economics to Ding Bogen, a Dutch economist who applied mathematics and statistical methods to economic analysis, there has been a worldwide upsurge of mathematicization of economic research. This tendentious atmosphere in economic research has had a great influence on China's economic theoretical circles. Some articles on economic theory lead to large paragraphs of mathematical formula derivation, and some academic economic magazines (not econometric or statistical magazines) even account for 1/2 to 2/3. Many economists have doubts about this: Is this the direction of economic theory research? Can such research solve or clarify some practical problems in China's economic system reform?

First, economic research is inseparable from mathematics.

The history of science reveals the fact that all disciplines belonging to "science" are based on the practice of human social activities. The division of disciplines and the induction of their respective characteristics are all the results of "human" factors. As far as the intrinsic nature is concerned, the interaction, mutual influence and mutual penetration between disciplines are extremely obvious, not only within natural science and social science, but also between the two disciplines.

Economics is a science that studies the allocation of social resources and socio-economic relations. Based on the measurability of resource stock and flow, in order to make resource allocation more fair and efficient, it is necessary for economics to use mathematics as a rigorous, accurate and practical thinking tool. The economic relationship formed in the process of resource allocation involves economic system, social psychology, values and other factors that are difficult to quantify. Economics, as an empirical science that focuses on speculation and qualitative analysis, cannot take mathematics as the basis or universal tool in economic research.

The role of mathematical methods in economics has been controversial in theoretical circles for at least 100 years. From "opposing the obscurantism of mathematics" to asserting that there is no science without mathematics, the views are quite different.

As a theoretical summary and abstraction of actual economic activities, economics has never left mathematics from its germination to its formation. On the one hand, the concept of number is produced in the long process of production activities. On the other hand, production activities always need different disciplines of economics, such as demography, marketing, labor and wages, price, finance, finance, accounting, etc. These disciplines are all related to counting, measurement and calculation. Without the concept of number and calculation method, it can be said that there would be no such subject.

The practice of economic activities determines that the study of economic theory is inseparable from quantity, and the application of mathematics in economics is closely related to the development of mathematics itself. Throughout the history of mathematics, it can be divided into four basic stages with qualitative differences. The first stage, the counting and arithmetic period (ending in the 5th century BC); In the second stage, elementary mathematics is a period in which mathematics remains unchanged (ending in17th century); The third stage, the period of variable mathematics (ending in19th century); The fourth stage, the period of modern mathematics. The outstanding characteristics of the modern mathematics period are that the branches of mathematics are constantly developing and expanding, the object and application scope of mathematics are greatly expanded, and the most common and unified concepts in mathematics are revealed with higher theoretical abstraction and generalization.

Although the concepts and conclusions of mathematics are extremely abstract, they all come from reality and can be widely used in other disciplines and social life practice, which may be the fundamental reason why mathematics not only has infinite vitality but also has great influence and attraction to all disciplines. As Engels said in Anti-Turin, the root of the possibility of applying mathematics to study the real world lies in that mathematics is extracted from the world itself and only shows the forming part of the internal relations in the world, so it can be widely used.

With the development of mathematics, the application scope of economics to mathematics is also expanding. /kloc-before the 0/9th century, elementary mathematics was mainly used in economics. From william petty's Tax Theory (1662) and Political Arithmetic (1676) to Quesnay's Economic Table (1758), the situation and changes of national wealth are described and analyzed with figures, charts and simple calculations. Since19th century, the concepts of variables and functions have been introduced into the study of economics, and the application of mathematical methods has become more common. Among them, Conrad's Research on Mathematical Principles of Wealth Theory (1838) is a book that consciously uses mathematical formulas to explain economic problems. After that, Tu Neng was based on the empirical formula of actual quantity (1850), Walras's equilibrium trading theory (1874), Harold's economic growth model (1948), Ding Bogen's 48-equation large-scale economic growth model (1939) and Lewis's "dual economy". Tobin's intermediate variable model (1958) and Solow's and Roman's economic growth models in the 1970s and 1990s. , published a large number of books on studying economic problems by mathematical methods. The common feature of these works is that they not only use general economic concepts and traditional economic methods, but also use the simplest mathematical symbols to the latest mathematical methods

From the inseparable development of economics and mathematics, we can know that mathematics can provide a unique and rigorous analysis method for economics. Like logic commonly used in qualitative analysis, mathematics is a tool to understand the world. However, the application of mathematics is meaningful only if it is combined with the profound theory of specific phenomena and the strict regulation of "quality", otherwise economic research will fall into the game of formulas and mathematics without substantive content.

Second, the deviation of using mathematical methods in economic research.

At present, the focus of debate about the application of mathematics in economic research is not whether economics should use mathematical methods, but how to use them. For the former, the practice of widely applying mathematics in economic activities and the research results of applying mathematical methods to economic theory have been answered positively, while for the latter, there are different views and opinions. It leads to serious deviation in the application of mathematical methods in economics, which affects the research effect. If it continues to develop, it may lead China's economic research astray.

The main problems in applying mathematical methods in economic research are:

1. The scope of application is too wide. The boundary of mathematical application is something that can be quantified, and the field of vision of economic research is all human economic activities and social relations. Not all economic activities and economic relations can be quantified, especially socio-economic relations, which are influenced by many social factors such as system, morality, culture and history, and these factors are almost impossible to quantify. It seems reasonable to use mathematical formulas to express the relationship between unquantifiable factors, because there is no operational relationship between them at all, and it is impossible to verify right or wrong by quantitative calculation. Although mathematics is also a language that reflects people's thinking, not all sciences can be transformed into the language of mathematics. The same is true of physics, chemistry, biology and other disciplines closely related to mathematics. Some problems may not be solved even if they are transformed into mathematical relations. However, social science, which studies human social activities, has more restrictions on the application of mathematics. Trying to dehumanize economics, even "mechanize" people in economic activities, and formalize people's activities is undoubtedly a kind of self-destruction of economic research.

It is easy for economics to indulge in the exploration of methodology and stick to microeconomic research, while ignoring and ignoring the overall problems involving macroeconomic system reform, mechanism design and social relations adjustment. As Richard Blencke said, modern economics is more and more keen on complex mathematical calculations, complacent about wonderful mathematical models and playing with mystery. The result is that economics is gradually divorced from the richness, complexity and irrationality of daily life. The trend of economic research in recent years has revealed some worrying signs in this regard.

2. The choice of mathematical model constraints is too arbitrary. Almost all theories are based on setting some premises and assumptions. For example, there are four accounting assumptions in accounting: accounting subject, going concern, accounting period and monetary measurement, while there are assumptions of "economic man" and "complete marketization" in western economics. The logical rigor of mathematical methods and the essence of calculation accuracy determine that any mathematical model is bound by several conditions, and only when these conditions are met can the mathematical model be established. The more complex the equation is, the more constraints it will be. At present, some economists completely ignore the constraints when establishing mathematical models, which is too simplistic, and the determination of constraints is very arbitrary, starting from the needs of the model itself, regardless of whether it meets the objective and practical requirements. The mathematical model thus established cannot quantitatively simulate economic phenomena and abstract economic theories. On the contrary, it is easy to cause theoretical confusion and major mistakes in practical operation.

3. The purpose of applying mathematical methods is not clear. Mathematics is also a language. The reason why some phenomena should be described by mathematics instead of other forms of language (such as words, pictures, music, limbs, etc.). ) because it can express this phenomenon more concisely and accurately than other forms of language. If concise and accurate results cannot be achieved, other language forms should be adopted. Some economists don't quite understand this. They deliberately use mathematical formulas that ordinary people can't understand to express problems that could have been explained in plain language, but their conclusions are common sense of general economics. It seems that the purpose of doing this can only be explained as follows: it can cover up the embarrassment of poor economic theory, it can save the labor of objective investigation, it can despise peers in the economic field with profound mathematical knowledge as capital, and it can practice the scholarship of "the so-called theory is to describe simple things in obscure language". In this regard, western economics also has many profound lessons. For example, in the 1990s, some economists tried to study financial problems with stochastic differential and nonparametric statistical methods, but so far, the results were minimal, and even fatal deviations appeared in the application.

4. In order to build the model consciously, we take a pragmatic attitude towards the actual data. Originally, to establish a mathematical model, it is necessary to conduct a detailed investigation of the phenomena studied, obtain as detailed digital data as possible, and conduct in-depth analysis from coarse to fine, from false to true, from this to that, from outside to inside, so as to find out the quantitative relationship between the main factors and each factor, and thus establish a mathematical expression. But now some economists do the opposite and reverse the order of establishing mathematical models. First, determine the mathematical expression, and then find the data that can support the establishment of mathematical relations, so as to verify the correctness of my theoretical summary. This subjective consciousness-oriented research method is not desirable. Seriously speaking, it is strongly idealistic. In fact, it has the same effect as computer fortune telling, although it is covered with the "scientific" coat of mathematics. Economics should be an empirical science that is constantly verified and enriched by practice from practice to theory to practice. If you do the opposite, it will inevitably lead economic research into the wrong path of not asking people's sufferings and staying away from the reality of social and economic life.

5. The effect of forecasting and analyzing economy with mathematical model is not ideal. Just taking stock price forecast as an example is enough to illustrate this point. The stock market can be said to be the experimental field with the most abundant and accurate information and the best conditions to fit the mathematical model according to various relevant data. People always try their best to establish various mathematical models to predict the trend of stock prices. At present, there are more than a dozen stock market analysis softwares on the market, such as Qianlong, Shenglong, Winner Star and Compass, but no matter which software is used to predict and analyze the stock trend, it seems that the odds of winning can only be maintained at about 50%. The inability to accurately predict the future trend is also the charm of the stock market to attract investment and speculation. Recently, some people who are engaged in theoretical physics research think that stock prices are also applicable to Heisenberg uncertainty principle in quantum physics. The operation of the whole macro-economy and economic problems such as price, unemployment and economic growth are much more complicated than the stock market. It is unrealistic to try to accurately analyze and predict its dynamic changes with one or two mathematical models, otherwise economics will fall into an embarrassing "chaotic" realm. The most famous example "butterfly effect" illustrates the limitation of mathematical model in practical application. Lorenz, a meteorologist at MIT, used a computer to solve the 13 equation which simulated the earth's atmosphere to predict the weather. In order to improve the accuracy of prediction, he took out a small intermediate variable. However, when he came back after drinking a cup of coffee, he was surprised to find that this small change made the difference between the results! There is nothing wrong with the computer, and his changes are reasonable. Why did it turn out to be heaven and earth? Lorenz thought hard and finally decided that he was caught in the "chaos" phenomenon: the extreme instability of the initial value led to the great difference in the final result. For example, a trivial butterfly in the Caribbean may just tease and vibrate its beautiful wings one day, and a few months later, there will be a powerful and unstoppable tornado on the earth! Chaos is everywhere. The universe is like this, the earth is like this, and so is the economic phenomenon. The mathematical model established by people can only show the integrity, generality and tendency of a phenomenon. Even the height and weight are highly correlated natural phenomena, and the regression equations fitted by statisticians and biologists all over the world are different, not to mention the socio-economic phenomenon mainly oriented by human thinking and behavior. In the past 200 years, most of the mathematical models that have stood the test of practice and been widely adopted by people in the history of economics are simple and can describe the general trend of things. Such as Engel coefficient, Gini coefficient, Rasbell index, Paishi index, Harold-Thomas economic growth model, Cobb-Douglas production function, Keynesian consumption function, Hicks IS-LM model, etc. Compared with the profusion of economic works, the number of such mathematical models is pitifully small, which makes people disappointed with the achievements of applied mathematical methods in economic research. As Lewis said in the book "Theory of Economic Growth", "Most predictions are not feasible in methods", "In order to predict what will happen, we have to know how all variables will change. It is impossible to build an equation system with thousands of variables, and we can predict the future by our own brains."

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