Linear Economic Models A. Brief Background Let us consider an economy consisting of n industries, each producing only one commodity needed by the others and possibly by itself. For example, suppose we have coal, steel, and auto, which are interrelated in a way described by a 3 x 3 matrix as follows. Let cij be the dollar amount of the i th commodity needed to produce 1 dollar's worth of the j th commodity. 0.20 0.45 Auto 0.05 0.15 0.10 It is seen from this matrix, that auto is the largest consumer of steel and steel is the largest consumer of coal.
Steel is most dependent on auto to survive.
This matrix is an example of an input-output , or a consumption matrix describing the interdependency of the economic sectors. The entries of such a matrix are nonnegative less than 1. In addition, the sum of the entries of each column should be less than 1 if each sector is to produce more than it consumes. He received the 1973 Nobel Prize in Economics for this work. Let xi be the total amount of output needed to be produced by the i th sector to satisfy the demands of all sectors.