<p>I found what I believe to be slightly different interetations of the individual elements of the Leontief Inverse (total requirements matrix):</p>
<p><a href="http://digamo.free.fr/io2009.pdf" rel="nofollow noreferrer">Miller and Blair (2009)</a> just after Equation (2.12) mention that for the Leontief Inverse <span class="math-container">$\mathbf{L}$</span></p>
<p><span class="math-container">$
(\mathbf{I}-\mathbf{A})^{-1}=\mathbf{L}=[l_{ij}]
$</span></p>
<p>elements <span class="math-container">$(i,j)$</span> can be represented as</p>
<p><span class="math-container">$
\partial x_i / \partial f_j = l_{ij}
$</span></p>
<p>where</p>
<p><span class="math-container">$x_i \dots$</span> output of sector <span class="math-container">$i$</span> <br />
<span class="math-container">$f_j \dots$</span> final demand for output of sector <span class="math-container">$j$</span></p>
<p>On the other hand, <a href="https://orbi.uliege.be/bitstream/2268/217602/1/Structural%20decomposition%20multipliers%20input-output.pdf" rel="nofollow noreferrer">Crama et al. (1984)</a> on the first page mention that <em>(variables changed to fit the convention used by Miller and Blair)</em></p>
<blockquote>
<p>Each element <span class="math-container">$(i, j)$</span> of <span class="math-container">$\mathbf{L}$</span> gives the variation in output of the sector <span class="math-container">$i$</span>, due to an increase of one unit of output, in the sector <span class="math-container">$j$</span>.</p>
</blockquote>
<p>While they do not provide an equation, this would be equivalent to (?)</p>
<p><span class="math-container">$
\partial x_i / \partial x_j = l_{ij}
$</span></p>
<p>Which is correct? Am I misinterpreting the description by Crama et al.?</p>
<p>I am aware of <a href="https://economics.stackexchange.com/q/23781/">this question</a> and <a href="https://math.stackexchange.com/q/139801/">this question</a>, but I believe neither provides sufficient information to answer my own.</p>

CC BY-SA 4.0
2023-03-30T10:55:44.443Z
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: https://strathprints.strath.ac.uk/61254/1/Swales_Turner_2017_Environmental_economics_handbook_of_input_output_analysis.pdf
EB3112
CC BY-SA 4.0
2023-03-30T14:02:47.993Z
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.
Wasserwaage
CC BY-SA 4.0
2023-03-31T12:54:31.840Z
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.
EB3112
<macroeconomics><international-trade><leontief><input-output>
Interpretation of the Elements of the Leontief Inverse (Total Requirements Matrix)
Wasserwaage
# Interpretation of the Elements of the Leontief Inverse (Total Requirements Matrix)

(Asked by: username_0 on 2023-03-30)

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)](http://digamo.free.fr/io2009.pdf) 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)](https://orbi.uliege.be/bitstream/2268/217602/1/Structural%20decomposition%20multipliers%20input-output.pdf) 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](https://economics.stackexchange.com/q/23781/) and [this question](https://math.stackexchange.com/q/139801/), but I believe neither provides sufficient information to answer my own.

**Tags:** macroeconomics, international-trade, leontief, input-output
> **Comments:**
> **username_1**: 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: https://strathprints.strath.ac.uk/61254/1/Swales_Turner_2017_Environmental_economics_handbook_of_input_output_analysis.pdf  
> **username_0**: 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.  
> **username_1**: 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.  

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## 0 answers

