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Mathematical economics is an area of **applied economics** that uses various principles, methods and tools of math to understand and analyze economic problems. In simple words, it is a field of study that relies on quantitative methods to explain economic phenomena. Mathematics and economics are often regarded as disciplines that are complementary to each other.

**Mathematical** and **statistical concepts** are extensively used in most areas of modern economics for quantitative analysis of problems. Methodical needs and problems of economics have also motivated many important areas of research in mathematics. Both disciplines have made important contributions to research and development of new concepts and methods in the other’s area.

For example, eminent economist **Kenneth Arrow** is well-known for his path-breaking work in the area of **mathematical optimization**. Mathematician **John Nash** on the other hand was awarded the 1994 Nobel Prize in economics for his contribution to the development of game theory which is now considered as central to modern economic theory.

The integration of mathematical and economic principles and statistical methods has led to the development of the essential set of analytical tools known as **econometrics**. The advancements in big data techniques, computing power and mathematical applications have also contributed significantly to quantitative methods becoming an integral part of modern economics.

Mathematical economics involves defining all relevant conditions, assumptions, and causal structures of economic theories in quantitative or mathematical terms. Doing this has two main benefits for economists. Firstly, it allows theorists to describe economic phenomena and make accurate inferences based on common definitions and assumptions of mathematical tools such as calculus and algebra. Secondly, putting theories into operation and inferences to empirical testing using real-world data helps them immensely in making quantitatively validated predictions for policymakers, investors, and firms.

Typically applied mathematical concepts and methods in economics include algebra, calculus, differential equations, matrices and determinants, mathematical programming, and computational methods. Economists claim that the application of these methods allows them to formulate precise and simple theoretical relationships. Mathematics allows them to create highly accurate and well-defined economic models which can be used to derive precise conclusions based on mathematical logic.

Logical, meaningful and testable propositions about a wide range of complex economic subjects which are difficult to express in informal terms can easily be formed using mathematics. The mathematical language is very useful for economists to make positive and specific claims on contentious and controversial subjects, which is again almost impossible to do informally. While research in economics is heavily influenced by the researcher’s bias, the application of mathematics in such definition and testing allows economists to reduce some of this bias.

Mathematical modeling also plays a crucial role in economic policymaking by assessing and forecasting the impact of different policies on economic activity. Research papers in economics are almost incomplete without quantitative methods and are rarely published without them.

The influence of mathematical economics on economic activity is such that decision makers, from government to businesses cannot imagine formulating policies without considering hard, accurate and quantitative predictions that it provides. For example, central banks rely on predictions about the impact of changes in interest rates on the GDP growth rate and inflation in formulating their monetary policies.

Broad areas of application of mathematical economics include:

- Solving optimization problems about goal equilibrium for policy makers, firms and individuals or households.
- Static analysis or equilibrium analysis in which the economic system (economy or market) and economic unit (firm, household, etc.) are modeled as unchangeable.
- Comparative statistics which studies the change from one equilibrium to another caused by changes in one or more relevant factors.
- Dynamic analysis which tracks the changes in an economic system for a given period of time.

The term econometrics was coined by the Norwegian economist **Ragnar Frisch** who also founded the journal called **Econometrica** in 1930. In a 1944 article in the journal the assertion that high-precision statistical analysis could be used to great result in validating mathematical concepts relating to economic actors by analyzing data from various complex sources.

Thus, econometrics is the integration of mathematical economics and statistical methods which helps economists in translating abstract theories into functional tools for developing economic policies, from the simplest to the highly complex. It aims to translate qualitative statements into meaningful quantitative data.

Econometrics is used in several branches of economics including **microeconomics**, **macroeconomics**, **labor economics**, finance and economic policy making. It is particularly useful for economists in solving problems of optimization where the objective is to determine the most effective modification from several alternatives to achieve a specific outcome. The flooding of data and information and the need to analyze them has made econometrics an essential part of economics. Within the broader discipline of economics, econometrics is used much more than mathematical economics for its wide range of statistical tools and methods.

The most used methods of statistical econometrics include time series analysis and linear regression which are applied to evaluate economic data.

A significant part of classical and modern economics can be represented though basic mathematical notation or geometrical terms. However, as noted earlier, mathematical economics typically uses more advanced methods such as calculus and matrices to make accurate predictions about economic phenomena that would be difficult to do without them or simply by using elementary math.

Using such advanced concepts becomes important because most economic problems involve so many variables that it is practically not possible to translate them into meaningful conclusions without their application.

Economist **Alfred Marshall** stated that every quantifiable, and analytically and statistically presentable should only be solved through mathematical work. Modern economics now depends almost entirely on mathematical methods and the tools employed for problem-solving have become increasingly sophisticated. This means that a command on mathematics has also become more important for economists than ever. This is the reason that undergraduate programs in economics require a strong mathematical foundation and go mostly to mathematicians.

The scope of mathematics in economics is such that many problems in economics integrate applied mathematics in their definition. This integration is based on the development of economic problems as elaborate models having clearly defined assumptions and predictions that can either be validated or falsified. The modeling can be both informal and simple, or formal, precise, and mathematical.

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