# Find the production matrix for the following input-output and demand matrices using the open model. A = [.3 .4] D = [4] [.35 .2] ...

Find the production matrix for the following input-output and demand matrices using the open model.

A = [.3 .4] D = [4]

[.35 .2] [6]

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### 1 Answer

Given input-output (technology) matrix `A=[[.3,.4],[.35,.2]]` and demand matrix `D=[[4],[6]]` , find the production matrix X:

`X=(I-A)^(-1)D` where I is the appropriate identity matrix (in this case `I=[[1,0],[0,1]]` ):

`I-A=[[1,0],[0,1]]-[[.3,.4],[.35,.2]]=[[.7,-.4],[-.35,.8]]`

Then `(I-A)^(-1)=1/((.7*.8)-(-.35*-.4))[[.8,.4],[.35,.7]]`

`=1/.42[[.8,.4],[.35,.7]]`

`=[[40/21,20/21],[5/6,5/3]]`

Now `(I-A)^(-1)D=[[40/21,20/21],[5/6,5/3]][[4],[6]]`

`=[[40/3],[40/3]]`

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The production matrix (vector) is `[[40/3],[40/3]]`

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