Chapter 4 - Nonnegative Square Matrices and Stability in Economic Systems

Abstract

This chapter discusses nonnegative square matrices and stability in economic systems. The purported Frobenius theorems regarding a nonnegative and nonzero square matrix are established by the utilization of elementary relations between submatrices. A square matrix is said to be decomposable if it can be partitioned by a permutation matrix. A permutation matrix is composed of all different unit vectors. The Frobenius root of an indecomposable nonnegative square matrix is a simple root. Nonnegative square matrices recur in Leontief's input–output models and the multiple markets theory of Hicks and Metzler. The Frobenius theorems are an important mathematical arsenal for these multi-sectoral analyses. The Frobenius root is the dominant root among all the roots. In the context of the Leontief system, inequalities have the implication that the increment in output of good resulting from an increase in the final demand for good is not larger if outputs of a number of goods are held constant than it would be if some of them are permitted to vary. However, the stability analysis of multiple markets in an n-good economy is when the supply of and demand for each good are equalized, when all the markets are cleared, and when the prices prevailing in this situation are the equilibrium prices.