Mathematical economics

Mathematical economics is the use of math to understand and solve problems in economics. It helps economists create clear descriptions of how money, resources, and choices work together. By using math, economists can test ideas and make predictions about complex subjects.

Math lets economists make specific statements about topics that might be debated. Many economic theories today are shown as mathematical models. These models are simplified versions of real situations. They help clarify what is being assumed and what results are expected.

Mathematical economics includes many kinds of problems, like finding the best way for a household or business to reach its goals, studying how markets stay balanced, and looking at how economies change over time. The use of math in economics began in the 1800s and grew a lot during the 20th century, especially after the Second World War, with the development of game theory. Some famous economists, like John Maynard Keynes, have questioned whether all human behavior can truly be described with math.

History

Main article: History of economic thought

People started using math to study economics in the 1600s. In Germany, they collected data to help the government. In England, they used numbers to study government problems, called Political Arithmetick.

In the 1800s, economics used even more math. Scholars made models to show how people act in an economy. One model was made by Johann Heinrich von Thünen in 1826. He studied how land is used for farming. Later, people who knew math from physical sciences started using those ideas in economics.

Marginalists and the roots of neoclassical economics

Main article: Marginalism

Scholars like Augustin Cournot and Léon Walras used math to show how people make choices. Cournot looked at competition between two sellers. Walras studied the whole economy. These ideas helped shape how we study economics today.

Modern mathematical economics

From the late 1930s, new math tools were used to make economic theory better. These tools helped economists explain economic behavior more clearly.

Vilfredo Pareto studied microeconomics by looking at how people make choices. Paul Samuelson used math to find patterns in economics in his book "Foundations of Economic Analysis" in 1947. He showed that economic systems can be modeled like systems in nature.

John von Neumann created math models of growing economies. Wassily Leontief developed ways to show how changes in one part of the economy affect other parts. Mathematical optimization helps find the best solutions to problems. Game theory, developed by John von Neumann and others, studies how people and groups make decisions when they work together. Agent-based computational economics uses computer simulations to study complex economic systems.

Mathematicization of economics

In the 20th century, many economics articles were written by university teachers. These articles often talked about ideas in economics, and they used more math over time. In 1892, most articles did not use hard math. But by 1990, very few articles did not use math. By 2003 and 2004, almost all articles in the best journals used math or numbers to explain their ideas.

Econometrics

Main article: Econometrics

Ragnar Frisch created the word "econometrics" and started the Econometric Society and the journal Econometrica. His student, Trygve Haavelmo, proved that careful statistics could check economic ideas using real data.

Henry L. Moore, an American economist, looked at farm work and tried to link crop amounts to math shapes. Even though his ways had mistakes, his work helped build economic models. After, Nicholas Kaldor made Moore’s ideas better with a model for business changes.

Application

Mathematical economics uses math, like calculus and matrix algebra, to study economic ideas. These tools help make hard economic problems easier to understand and solve. Many economic ideas are best shown with math because they have many parts and connections.

Economists today use math a lot, and students studying economics usually need to be good at math. This has led many mathematicians to work in economics, using their skills to solve real-world problems. Economic models can be stochastic, meaning they include random events, or deterministic, meaning they follow fixed rules.

Discussions of validity

Different groups of economists have different ideas about using math in economics. Some, like the Austrian school, think that economics can't always be shown with numbers and math. They believe it makes economics seem more like a science than it really is.

Other economists, like Paul Samuelson and Robert M. Solow, think that math is very useful in economics. They say it helps to explain tricky ideas clearly and has helped us learn more about economics.