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# A primitive economy depends on two basic goods, yams and pork.  Production of 1 bushel...

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A primitive economy depends on two basic goods, yams and pork.  Production of 1 bushel of yams requires 1/4 bushels of yams and 1/2 of a pig.  To produce 1 pig requires 1/3 bushel of yams.  Find the amount of each commodity that should be produced to get 300 bushels of yams and 50 pigs.

Posted by cbromm on July 17, 2013 at 12:39 AM via web and tagged with math, matrix

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Let Sector 1 be yams and Sector 2 be pigs.

The external demand vector D is the required amount to produce which is `D=[[300],[50]]` .

The technology matrix A gives the internal demand required to produce 1 unit in each sector. `a_(11)` is the requirement from sector 1 to produce 1 unit in sector 1, `a_12` is the requirement from sector 1 to produce 1 unit from sector 2, etc...

So `A=[[1/4,1/3],[1/2,0]]`

We are looking for the production vector X where `X=(I-A)^(-1)D`

`X=[[3/4,-1/3],[-1/2,1]]^(-1)[[300],[50]]`

`=[[12/7,4/7],[6/7,9/7]][[300],[50]]`

`=[[3900/7],[2250/7]]`

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Thus we need approximately 558 bushels of yams and 322 pigs.

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Posted by embizze on July 17, 2013 at 2:31 AM (Answer #1)