On the Role of Linear Algebra in Microeconomics

Abstract

Microeconomics has a long history, and it is originated in the Adam Smith's insight,which is so called "invisible hand of God".However, it took much time to formulate and analyze microeconomic models explicitly. This is because it was very difficult to realize the market mechanism in which the prices of goods and the supply and demand control each other automatically. One of the most important break-through was the idea of P.A.Samuelson. He formulated the process of controlling prices and (excessive) demands as a sort of self-excited systems. Based on the models like this,various properties of market mechanism such as the efficiency,the existence, the uniqueness and the stability of general equilibium have been clarified in terms of mathematics (including analysis in many variables,dynamical systems and differential topology) in the letter half of 20th century. This paper is aimed to generalize the sufficient conditions to guarantee the local stability of general equilibrium in case of pure exchange economy (with no production or consumption). Through the arguments in this paper,the importance of linear algebra in microeconomics should be illustrated.