If not the whole output but only a portion of it is consumed by the industries themselves,

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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

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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

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where αjk is the fraction of the output of industry k consumed (purchased) by industry j. Let Pj 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.

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