Question

Suppose that three industries are interrelated so that their outputs are used as inputs by themselves, according to the 3x3 consumption matrix

A= [ajk]= [0.1 0.5    0 ]
                | 0.8   0    0.4 |
                [0.1   0.5   0.6]  

where, ajk is the fraction of the output of industry k consumed 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], and find a solution p with nonnegative p1, p2, p3. Also, show that the consumption matrix A must have column sums 1 and always has eigenvalue 1.

Answer 1

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