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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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    Viktor has answered your question, but I thought I would add that you are looking for what's called an *eigenvector* for your matrix $A$.2011-03-18
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    @Jasen: If you have a new question, you shouldn't just edit your previous one, you should open a new one.2011-03-18
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    @Jasen: I have reverted your question to its original form. If you want to ask a new question, start a new thread.2011-03-18
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    @Jasen: By the way, [this question](http://math.stackexchange.com/questions/16151/how-can-a-x-mod-m-have-multiple-meanings-in-modular-arithmetic) might resolve your difficulties with modulus notation.2011-03-18
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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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