# Interpretation of the Elements of the Leontief Inverse (Total Requirements Matrix)

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

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

where

$$x_i \dots$$ 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.

• 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/… Mar 30, 2023 at 10:55
• 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. Mar 30, 2023 at 14:02
• 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. Mar 31, 2023 at 12:54