What does the Leontief Inverse represent? (Intuitive Meaning or Real World Concept)

Question

I encountered that nobody, even my profs and lecturers so far, has an intuitive way to explain what the Leontief inverse represents. Does somebody here?

As most people here would know, it goes:

$\ x=Ax+y $ where $x$ is the output and $y$ the final demand. So

$\ x - Ax =y $

$\ x (1 - A) =y $ or

$\ x = (1-A)^{-1} y $ or

$\ x = Ly $

While $A$ is clearly the proportion of total output that comes from intermediate value exchanges and not from transactions to final demand $1 - A$ should be the proportion of total output caused by final demand. However $L = (1-A)^{-1}$. So what would this be? The inverse proportion of what in total output is related to final demand? You can also just say of course $L$ represents the multiplier or scaling factor of how $x$ responds to changes in $y$, but that is not what I am asking for. I wonder if there is any real world interpretation possible on what L represents. Or is it an entirely abstract multiplier?

Answer 1

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.

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Under the condition that $1>|A|\geq0$

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

Which allows us to write

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

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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

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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.

Answer 2

What intuition can the mathematical concept of an inverse (function, matrix) have?

In a single-input monotonic production function $F(x) = q \implies x= F^{-1}(q)$ the operator $F^{-1}$ is the transformation mechanism that translates output to required input. A "change of units" calculator if you wish.

Isn't this what $(I-A)^{-1}$ does in the case of the OP's question? Given whatever technical, or behavioral, or other assumptions we make (here, about how much is needed for intermediate use, and how much is demand), $(I-A)^{-1}$ translates the units of output into units of input.

But it appears the OP already knows that much, and they are asking for something more...

How about this then: $(I-A)^{-1}$ compacts nicely the vast number of workhours, committees, memos, disagreements, power struggles etc, that would be needed in a planning economy, where we would have to secure amount $x$ of the input in order to satisfy amount $y$ of the output.

I leave as food for thought what could it represent in a market economy, where prices and decentralized decisions enter the picture.