( Leontief input-output model ) Suppose that three industries are interrelated so that their outputs are used as inputs by themselves, according to the $3 \times 3$ consumption matrix
A = [$a_{jk}$] = $ \left[ \begin{array}{ccc} 0.1&0.5&0\\ 0.8&0&0.4\\ 0.1&0.5&0.6 \end{array} \right] $
where $a_{jk}$ 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. Determine that there is a price vector such that $~~~~$ p = $[~~p_{1} ~~~ p_{2} ~~~ p_{3}~~]^{T}$ $~$ for this scenario.
Any ideas on how to go about solving this??
Thank You in advance.
$Income = Expenditures$ condition would give you $A\,\vec p=\vec p$ or $(I-A)\vec p=0$. Such homogeneous SLE will have infinitely many solutions (a parametric family): $$\vec p=c\left[\begin{array}{c} p_1 \\ p_2 \\ p_3 \end{array}\right]$$
Please see: Elementary Linear Algebra: Applications Version, Howard Anton, Chris Rorres. - 10.8 Leontief Economic Models