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To analyze problems in economics and form theories, mathematical economics is used which rests on different mathematical methods. The mathematical methods employed to obtain accurate results include certain complex mathematical operations like differential and integral calculus, differential equations, mathematical programming, and complex algebra. They cannot be left out in any economic analysis since they became an integral part of the study of economics.
Economic experts use mathematic calculations to develop test theories, make projections on complex and compound issues which would be hard to analyze without the math pattern. Precision and accuracy are the two major factors of mathematics, and they enable economic specialists to make precise claims on different matters. The analysts make their predictions based on economic models which represent different mathematical relations between different factors which help in clarifying assumptions and implications.
Broad applications that are mostly used are as follows:
Mathematical optimization (or computer programming) is used to solve problems at different levels: household, company, or governmental level. Comparative statics includes making changes to one or more factors and observe the difference. Dynamic analysis can be used to examine the development of an economy based on, for example, economic growth.
Overview of History
Economic modeling goes back to the 19th century when mathematic formulas were used to identify economic behavior through mathematical optimization use. From then on, economics has been linked to economic analyses, but it was only in the 1940s that mathematic formulations became widely used in economics. This means that mathematical economics is yet rather a more recent discovery. The use of mathematics to explain behavior was also strongly criticized since many considered that human behavior is unpredictable and cannot be summed up in a mathematic formula.
The earliest records of mathematics as an analyst tool in economy date back to the 17th century when the public administration was displayed through statistics and how it affected society. The study was conducted by a German University, which also produced the famous expert on these matters Gottfried Achenwall, who was the founder of statistics. Similar studies around the same time were also conducted in England, and it all related to policy making and the government and how it reflects on the public.
But maths needed two more centuries to enter economics completely, but it needed some more time to start using sophisticated mathematical operations. Thünen was an economist that sort of revolutionized the whole analysis process by developing new economic models to existing problems, rather than relying on former models which sometimes were not quite adaptable to new problems.
The introduction of calculus and linear models
Calculus was introduced in the 20th century, and it studies continuous changes in society and economy. For example, calculus can be used to determine the interrelation between education, income, and work experience. Marginal changes are often associated with calculus, and they refer to all the small changes that can affect a variable which in return affect yet another variable. The method helps business to determine the ratio between marginal revenues and marginal costs which further help them to increase profits.
The linear model is based on a formula which gives accurate calculations of the different economic variables that used in an analysis. The equation reads as pT (A - λ B) q = 0. P is the transposed probability vector and indicates the price of the goods, and q is the probability vector that stands for intensity of the production process (mass-process, etc.). The λ stands for the economic growth and is equal to the interest rate. Such linear programming made it possible to prove an increasing growth rate which always equals the interest rate.
The In-put and Out-put Process
This model comes from the Soviets who created a model for measuring production and demand developments, whereby the outstanding economist Leontief proved that changes in one economic sector, i.e. change in demand, automatically have an impact on production in a completely other economic sector. Using this model, outputs are calculated or estimated by using input coefficients and the price of inputs does not even matter.
This model refers to using the most perspective or highest in quality element from a number of offered parameters. In order to reach the optimum, certain general conditions have to be met, whereby different computational optimization techniques are used. Optimization can also be used for testing against empirical data.
Linear and Non-linear Programming
Linear programming was introduced in the 1930's in order to optimize the distribution of resources in bigger companies, and this technique was used during the Berlin blockade of 1948 to plan distribution of supplies by air to Berliners. Linear programming refers to getting the most out of the least, e.g. get the best outcome out of the minimum. Non-linear programming relates to the equality and non-equality systems of optimization problems and unknown real-life variables, whereby some of the variables are non-linear.
Econometrics is the study that unites mathematics and statistics for advancements in economics. Econometrics has rather been associated with statistics through the past. Statistical Econometrics is a branch of economics that uses linear regression and analyses time sequences to produce economic data. Econometrics was introduced in the 1930's, and it got its name from Ragnar Frisch, a great economic expert from that period. One of his students advocated that accurate statistical calculations can be a means of proof of mathematical theories.
Economics nowadays depends heavily on mathematical models and tools, as well as statistical analyses which became more and more sophisticated so as to be an integral part of economics. Professionals in economics and finances know that the essence of all predictions is mathematical methods.
Many college courses for economics include extensive courses in mathematics in order to produce better economic analysts who are taught to apply their mathematical knowledge to economic projections practically. The study of mathematics and its application to statistics and economics reduced errors in financial analyses to a tremendous extent. Economics can barely be separated from linear programming, algebra, or geometry, all of which represent the fundamental factors in achieving high-quality results from economic predictions and analyses.