If not the whole output but only a portion of it is consumed by the industries themselves, then instead of Ax = x (as in Prob. 13), we have x - Ax = y, where x = [x₁    x₂    x₃]^T is produced, Ax is consumed by the industries, and, thus, y is the net production available for other consumers. Find for what production x a given demand vector y = [0.1    0.3    0.1]^T can be achieved if the consumption matrix is

Data from Prob 13

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

where α_jk is the fraction of the output of industry k consumed (purchased) by industry j. Let P_j be the price charged by industry j for its total output. A problem is to find prices so that for each industry, total expenditures equal total income. Show that this leads to Ap = p, where p = [p₁   p₂    p₃]^T, and find a solution p with non negative p₁, p₂,p₃.