I found what I believe to be slightly different interetations of the individual elements of the Leontief Inverse (total requirements matrix):

Miller and Blair (2009) just after Equation (2.12) mention that for the Leontief Inverse $\mathbf{L}$

$ (\mathbf{I}-\mathbf{A})^{-1}=\mathbf{L}=[l_{ij}] $

elements $(i,j)$ can be represented as

$ \partial x_i / \partial f_j = l_{ij} $


$x_i \dots$ output of sector $i$
$f_j \dots$ final demand for output of sector $j$

On the other hand, Crama et al. (1984) on the first page mention that (variables changed to fit the convention used by Miller and Blair)

Each element $(i, j)$ of $\mathbf{L}$ gives the variation in output of the sector $i$, due to an increase of one unit of output, in the sector $j$.

While they do not provide an equation, this would be equivalent to (?)

$ \partial x_i / \partial x_j = l_{ij} $

Which is correct? Am I misinterpreting the description by Crama et al.?

I am aware of this question and this question, but I believe neither provides sufficient information to answer my own.

  • $\begingroup$ Hi @Wasserwaage. Demand-driven IO analysis has a passive supply side and fixed prices. The equilibrium process in demand-driven IOA is based upon supply quantities responding to exogenous demand shocks. The Leontief inverse thus solves for the necessary supply/input changes given a demand shock to keep the process in equilibrium. Section 2: strathprints.strath.ac.uk/61254/1/… $\endgroup$
    – EB3112
    Mar 30, 2023 at 10:55
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    $\begingroup$ I think both sources I found use the demand-driven convention. From what I understand from Section 2 of your document, it supports the definition used by Miller and Blair. Unfortunately, I don't see how it answers my question. $\endgroup$ Mar 30, 2023 at 14:02
  • $\begingroup$ The next breath is of course: elements measure the portion/share of inputs to sector J, which comes from sector I, and thus the corresponding change in these portions given a demand shock. If I am missing something, I know longer know how to help. Your title is better worded than the question, must say. Someone will provide a superior answer to me, but I think the Q needs more focus. $\endgroup$
    – EB3112
    Mar 31, 2023 at 12:54


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