3 Suppose that three industries are interrelated so that the

3) Suppose that three industries are interrelated so that their outputs are used as inputs by themselves, according to the 3 × 3 consumption matrix

where a_ij is the fraction of the output of industry j consumed (purchased) by industry i. Let p_i be the price charged by industry i for its total output. A problem is to find prices so that for each industry, total expenditures equal total income. This leads to the system of linear equations Ap = p, where p = [p₁, p₂, p₃]^T . Find a solution p with nonnegative p₁, p₂ and p₃.

Solution

Given matrix is

we need to find p such that Ap=p, where p=[p_1, p_2, p₃]

Ap=p implies [A-I]p=O, where O is a zero matrix o=[0,0,0]. Now A-I=

determterminant of A-I = (-0.9)(0.4-0.2)-(.05)(-0.32-.014)

= -0.9*0.2-0.5*(-0.18)

=-0.18+0.90

=0.72, therefore the matrix A-I is invertible

which implies that that the system of equation [A-I]p=O, has only trivial solution i.e,

P=[0,0,0] i.e p₁=0, p₂=0, p₃=0.