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( 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.

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    @Jasen: If you like Viktor's answer, it would also be good form to upvote it (click on the up arrow by the answer) and then to formally accept it (click on the check mark by the answer).2011-03-18

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$Income = Expenditures$ 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

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    @Jasen: The Google books version of the text Viktor cites is here: http://books.google.com/books?id=1PJ-WHepeBsC&pg=PA582&lpg=PA582&dq=Elementary+Linear+Algebra+%22Three+homeowners+-+a+carpenter,+an+electrician,+and+a+plumber&source=bl&ots=KdH4rt3v_8&sig=A6WWeyfNAV1rbJYVSuHXAr3LzTQ&hl=en&ei=cMKDTYqmBZT2tgOXs5X5AQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBYQ6AEwAA#v=onepage&q&f=false2011-03-18