Analysis on the Application of Mathematics in Economics
Abstract: For more than half a century, the application of mathematical forms in the field of economics has been an important development trend, which has also had an important impact on economic theory and practice. The popularization of western economics knowledge also makes mathematical knowledge permeate all aspects of economics. Looking through the famous journals of economics, we will find that the application of a large number of mathematical methods tends to surpass the students majoring in mathematics. The quality of economics papers depends on the application of mathematical methods, and the admission of masters and doctors in economics depends on their profound mathematical skills, and mathematics almost has the trend of unifying economics. What kind of rational beauty will economics deduce when it meets mathematics?
Keywords: economics; Mathematics; western economics
First, the definition of economics
The finiteness of resources and the infinity of human desires are the basis of the birth of economics, which is a simple but profound truth that ordinary people know. What economics needs to solve is actually a question of how to choose, that is to say, economics needs to solve the problem of how to allocate limited resources reasonably and effectively to meet human endless desires. Therefore, in western economics, economics is defined as a discipline that studies the allocation of scarce resources. It takes rational hypothesis as the logical starting point of studying human behavior. These problems based on realistic basic research are closely related to the problems existing in real economic life, and the conclusions of the research are helpful to explain or understand the real economic problems. However, the ultimate goal of economics to pay attention to human behavior itself is to pursue the efficiency of resource allocation.
Economics, as a discipline that studies the facts of human society, has its unique flavor. It can involve politics, society and other disciplines. For an economist, when he tries to explain the world, he is an economist, and when he tries to change the world, he is a politician. The special dual identity also shows the pluralism of economics. Some people even put forward a view that there is no difference between economics and historiography in essence, but historiography studies mostly the past, while the historical length of economics is not that long, and economics mostly uses mathematics and statistical tools to explain problems.
Second, the application of mathematics in economics
Western economists have introduced mathematics into economics in a large number, that is, they tried to explain the world in an accurate way, and then tried to develop modern western economics into an accurate science. Use Gao Hongye's "Western Economics (Micro-editing) Fourth Edition" >: For example, the application of limit and derivative in explaining marginal utility; Lagrange multiplier method used for cobweb model analysis: multivariate function differential method to prove the marginal rate of technology substitution: game theory and equilibrium concepts are used when expounding the game strategy between oligopoly manufacturers; And the ubiquitous application of various function curves and the derivation of function expressions. And these are just some examples in the introductory textbook of economics. In the whole field of economics, marginal analysis, Walras's general equilibrium theory, linear programming, input-output analysis, game theory, stochastic mathematics, fuzzy mathematics and nonlinear science are also widely used in economics. These tools, which originally belonged to mathematics, are now full of all aspects of economic research. At the same time, the establishment of the Nobel Prize in Economics seems to be a strong proof.
But we can't deny that mathematics, as a tool, also plays an important role in the interpretation of economic theory. Let's look at some classic examples.
1. marginal utility theory
/kloc-in the 0/7th century, with the disintegration of feudal society in Europe and the transition from capitalist workshop handicraft industry to machine mass production, a series of new problems that must be studied from the viewpoint of movement change and development were put forward to mathematics. Therefore, variable mathematics, a mathematical part that describes the law of movement and change of things in a certain quantity, came into being. From 65438 to the early 1970s, three scholars of different nationalities, jevons, Meng Le and Walras, combined their concepts of "desire" or "utility" with the basic concept of "differentiation", resulting in "marginal utility". The famous "marginal revolution" in the history of economics also broke out with the infiltration of calculus into economics. After the heyday of marginal revolution, the marginal analysis method itself developed in a deeper and broader direction. Marginal analysis, a powerful tool born out of calculus, is also widely used in various research fields of economics-macroeconomics, linear programming analysis, econometrics, welfare economics and so on.
2. General equilibrium theory
From 65438 to 2008, free competition capitalism was on the rise in Europe. Economists have noticed that there are many consumers and producers in a society, and their independent decisions have not caused confusion, but actually produced an optimal economic state. 1776, Adam Smith wrote in his economics bible.
Third, mathematical methods are tools in economics.
From the above examples, we can see that the flexible application of mathematics really plays a great role in the elaboration of an economic theory. But we must see that for economic theory, mathematical methods are tools for analysis, demonstration and research. Whether this tool can produce useful results depends on whether the economic theory of applied mathematics is correct. Mathematical methods can serve both the correct theory and the wrong theory. The equation is proved to be right, but the formula is right, but the content may be wrong. Mathematical equations are very useful, but mathematics itself has no content. Generally speaking, the world is so complicated and there are a lot of statistical traps, so it is better to draw a correct conclusion first. Therefore, the application of mathematics in economics should be carefully attached with conditions, which is by no means a problem that everyone can use if he wants to use it.
I remember Professor Minglu from Fudan University said in an article that originated from the relationship between economics and mathematics: "Intuition is very important in economics. After you have intuition, before you make a mathematical model, you should have a story and logic in your mind and write it down mathematically. Mathematics can really help you get some conclusions, but my experience tells me that 70% or even 80% of the conclusions may be known before you write mathematics; There is really a conclusion of 20% to 30%. If you don't write math, you may not know it, or your knowledge is vague. Why do I say that? Looking back at the starting point just mentioned, if you believe that relying on mathematics alone can help you explain economics clearly, then I want to ask, where did you start? When you write your mathematical assumptions, when you assume people's behavior and decision-making patterns, when you assume the market structure in the model-is it a monopoly market structure or a perfectly competitive market structure? Do you want to put the government in your model? In fact, what you have to do is to express your understanding of economic reality with mathematics. If you say that I don't understand this reality and write mathematics directly, a very dangerous result is that your starting point is wrong, so your conclusion can't be right, even if you are very fancy in mathematics. " In addition, Professor Minglu also emphasized the problem of "after mathematics". He said, "after you have deduced mathematics, have you ever thought about what its story is and what the economic significance behind mathematical logic is?" This point is often overlooked by students. In the process of studying and reading papers, if we ignore this point, we will only learn mathematics but not economics. When you are writing your thesis, you have finished writing math. You can write the word "certificate completed". You can finish 50% of your thesis at most. You know, at the mathematical level, as long as you move-bark and make small assumptions, you may get different conclusions. Therefore, the mathematical conclusion that exists out of the economic mechanism is meaningless. "
Therefore, thinking should be the most important thing, and mathematics is a tool. The purpose is to see the problem clearly and draw conclusions. Mathematical tools in economics are very important-just as important as communicating with foreigners in English. However, just like communicating with foreigners in English, what is more important is what you want to communicate. In economics, mathematics is a language that economists all over the world can understand. Similarly, a good language does not necessarily mean that your thoughts are profound. In the current school of economics, the new institutional economics, which does not use mathematics very much, is very explanatory. As a great economist in economic history, Nash, as a doctoral student in the Department of Mathematics, won the Nobel Prize in Economics in 199 1 for his pioneering contribution to game theory.
Did Nash win the prize only by mathematics? This is a mathematical dialysis idea, a groundbreaking idea. Coase, who never used mathematics, won the Nobel Prize in Economics only by the Nature of Enterprises published in his twenties and the Federal Communications Commission published later, and became a pivotal figure in economic history. Coase's property right theory and transaction cost theory prove the importance of property right system to economy, and on this basis, a very important new institutional economics school in economics is formed. Coase did not rely on any mathematical tools, but relied entirely on an idea, an idea pioneered by predecessors. There are also some economists who oppose the use of mathematical tools in economics. For example, Mirdal, who won the Nobel Prize in Economics in 1974, is an economist who speaks on behalf of the disadvantaged groups. His concern for African Americans and people in developing countries is the embodiment of humanistic care in economics. Hayek, an economist who won the prize in the same year, is a master of liberalism, and his exposition of freedom is undoubtedly the greatest concern for mankind.