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...

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.

Asked on by cbromm

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embizze | High School Teacher | (Level 1) Educator Emeritus

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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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