Leontief Inverse Matrix

Definition

Leontief Inverse Matrix (or Total Requirements Matrix) is a central mathematical object in Input-Output Analysis. The matrix represents the amount of gross output from sector i that is produced to satisfy a unit of Final Demand y from sector j.

Formula

The columns of the Leontief inverse (input-output) table show the input requirements, both direct and indirect, on all other producers, generated by one unit of output^[1]. The following equation shows the reduced form of the Leontief equation, which serves as the basis for analyses conducted with the basic static IO model:

\begin{align} x_{i} & = \sum_j a_{ij} x_{j} + y_j \\ x & = (I - A)^{-1} y \end{align}

where

$a_{ij}$ are the input coefficients for intermediate inputs,
$(I - A)^{-1}$ is the Leontief Inverse (denoted as $L$ ),
$y_{j}$ is the final demand vector
$x_j$ is the output of sector j (production value).

Usage

The allocation of production factors (e.g., employees) to final demand is calculated using the Leontief inverse, and gives the total production output (including upstream) involved in producing the company’s output.

All components of this equation can be derived from an IO table. The resulting set of linear equations (the Leontief quantity model or the demand pull model) can be used to analyse the impact of a company’s production on the economy-wide output and various indicators linked to that output. The results show the impacts of satisfying the company’s demand on the sectors of the economy and provide insights into the industry-wide direct and indirect effects.

Further Resources

References

Eurostat SUT Manual

↑ R.E. Miller and P.D. Blair, Input-Output Analysis: Foundations and Extensions, Second Edition, Cambridge University Press, 2009

[1] R.E. Miller and P.D. Blair, Input-Output Analysis: Foundations and Extensions, Second Edition, Cambridge University Press, 2009

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