A simplified economy involves just three commodity categories - agriculture, manufacturing, and transporation, all in appropriate units.  Production of 1 unit of agriculture requires 1/5 unit of...

A simplified economy involves just three commodity categories - agriculture, manufacturing, and transporation, all in appropriate units.  Production of 1 unit of agriculture requires 1/5 unit of manufacturing and 1/4 unit of transportation; production of 1 unit of manufacturing requires 1/4 unit of agriculture and 1/4 unit of transportation; and production of 1 unit of transportation requires 1/3 unit of agriculture and 1/4 unit of manufacturing.  If the demand is 610 units of each commodity, how many units of each commodity should be produced?

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Given an input-output matrix A, production matrix X and demand matrix D, they are related by the matrix equation

`D=X-AX`   

To find the production matrix, we need to solve for X

`D=(I-A)X`  where I is the identity

`X=(I-A)^{-1}D`

In this case, we have

`A=[[ 0 , 1/4 , 1/3], [1/5, 0, 1/4], [1/4, 1/4, 0]]`

Substituting into the formula for the production matrix, we get

`X=[[1,-1/4,-1/3],[-1/5, 1, -1/4],[-1/4, -1/4,1]]^{-1} [610,610,610]`

which can be solved to get

`X=[1317.14, 1182.13, 1234.82]`

There needs to be 1317.14 units of agriculture, 1182.13 units of manufacturing and 1234.82 units of transportation.

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