Economics Stack Exchange Archive

Understanding the Leontief inverse

• posted by: Seb on 2012-01-04
• tagged: macroeconomics, theory
• score: 2

What I remember from economics about input/output analysis is that it basically analyses the interdependencies between business sectors and demand. If we use matrices we have $A$ as the input output matrix, $I$ as an identity matrix and $d$ as final demand. In order to find the final input $x$ we may solve the Leontief Inverse:

$$x = (I-A)^{-1}\cdot d$$

So here’s my question: Is there a simple rationale behind this inverse? Especially when considering the form:

$$(I-A)^{-1} = I-A + A^2 + A^3\ldots$$

What happens if we change an element $a_{i,j}$ in $A$? How is this transmitted within the system? And is there decent literature about this behaviour around? Thank you very much for your help!

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