An economy in which the input requirements for production are directly proportional to the levels of production can be described by a set of linear equations. For details see Input Output Analysis. The linear equations can be expressed in terms of matrices.

Suppose an economy has n industries each producing a single unique product. (There is a generalization of input output analysis, called activity analysis, in which an industry may produce more than one product, some of which could be pollutants.) Let the product input requirements per unit of product output be expressed as an nxn matrix A. Let X be the n dimensional vector of outputs and F the n dimensional vector of final demands. The amounts of production used up in producing output X is AX. This is called the intermediary demand. The total demand is thus AX+F. The supply of products is just the vector X. For an equilibrium between supply and demand the following equations must be satisfied.

X = AX + F

The equilibium production is then given by

X = (I−A)^-1F

A viable economy is one in which any vector of nonnegative final demand induces a vector of nonnegative industrial productions. In order for this to be true the elements of (I−A)^-1 must all be positive. For this to be true (I−A) has to satisfy certain coditions.

A minor of a matrix is the value of a determinant. The principal leading minors of an nxn matrix are evaluated on what is left after the last m rows and columes are deleted, where m runs from (n-1) down to 0.

The condition for the nxn matrix of (I−A) to have an inverse of nonnegative elements is that its principal leading minors be positive. This is known as the Hawkins-Simon conditions.

History

(To be continued.)