What does the Leontief Inverse represent? (Intuitive Meaning or Real World Concept)
Something you didn't mention is that

$x = Ax + y$

means that to produce one unit of $x$, you use $A$ unit of $x$. E.g. you need electricity to produce electricity.

Under the condition that $1&gt;|A|\geq0$

$(1-A)^{-1} = \sum_{k=0}^\infty A^k$ 

Which allows us to write

$x = Ly = \left(\sum_{k=0}^\infty A^k \right)y$


Thus, for one unit of $y$, your (scalar) $L$ expressed as an infinite sum can be interpreted as the length of the engaged auto-production chain.

A little illustration




Note that all of the above can be generalized in matrix terms, i.e.

$\boldsymbol{x} = \left(\boldsymbol{I} - \boldsymbol{A}\right)^{-1}\boldsymbol{y}$

where $\boldsymbol{x}$ and $\boldsymbol{y}$ are column vectors and $\boldsymbol{I}$ and $\boldsymbol{A}$ square matrices. Each $\boldsymbol{x}$'s component may (or may not) require some of its own production as well as some units coming from other sectors, i.e. from $\boldsymbol{A}\boldsymbol{x}$. In this case, 

$\left(\boldsymbol{I} - \boldsymbol{A}\right)^{-1} = \sum_{k=0}^\infty \boldsymbol{A}^k$ is true if and only if all of $\boldsymbol{A}$'s eigenvalues have absolute value smaller than $1$. $\left(\boldsymbol{I} - \boldsymbol{A}\right)^{-1}$ can be useful if you want to know how any variation -- of $y$ admitedly, but also of anything else, say, a tax -- propagates through a (co-dependent) sectorial chain.