The Russian-born American economist Wassily Leontief (1906-1999) used matrix algebra and subsequently computer technology to trace the relationship between industries of the US economy.

Every output in an economy can be analyzed in terms of final consumption and all the inputs necessary for its production.

By developing a matrix of various production functions, it is possible to track the knock-on effect of a change in demand for a given good or service on other industries or sectors. Lower automobile sales might mean a decline in steel demand, a fall in electricity or a drop in coal imports.

Source:

W W Leontief, The Structure of the American Economy, 1919-29 (New York, 1941)

## Introduction[edit]

In recognition of the increasing importance of global resource use mediated by international trade for environmental accounting and policy, new perspectives have been and are currently being developed within environmental accounting. The most prominent among these are consumption-based accounts compiled using environmentally extended input-output analysis. Consumption-based indicators of material use are commonly referred to as “material footprints”^{[1]} (comparable to carbon footprints and water footprints) or as raw material equivalents (RME) for imported and exported goods.^{[2]}^{[3]} Raw material equivalents or material footprints of traded goods comprise the material inputs required along the entire supply chain associated with their production. This includes both direct and indirect flows: For example, the ore mined to extract the metal contained in a mobile phone as well as the coal needed to generate the electricity needed to produce the metal concentrates would be included. In order to allocate domestic extraction to exported goods, information on the production and trade structure of an economy is required. In monetary terms, information on the production structure is contained in commonly available economy-wide input-output tables (IOT) which recently have been combined with trade statistics to form multi-regional IO (MRIO) tables.

## Input-Output Analysis for EEIOA[edit]

In the following, a short introduction to input-output analysis and its environmental extension for the calculation of material footprints or RME indicators is provided. The inter-industry flows within an economy form an `n`×`n` matrix * Z* and the total output of each industry forms an

`n`×1 vector

*. By dividing each flow into an industry (i.e., each element of*

**x***) by the total output of that same industry, we obtain an*

**Z**`n`×

`n`matrix of so-called technical coefficients

*. In matrix algebra, this reads as follows:*

**A**{\displaystyle A=Z\times {\hat {x}}^{-1}}

where:

- {\displaystyle {\hat {x}}} represents the vector
diagonlized into a matrix ({\displaystyle {\hat {x}}=I{\vec {x}}})**x**

Matrix * A* contains the multipliers for the inter-industry inputs required to supply one unit of industry output. A certain total economic output

*is required to satisfy a given level of final demand*

**x***. This final demand may be domestic (for private households as well as the public sector) or foreign (exports) and can be written as an*

**y**`n`×1 vector. When this vector of final demand

*is multiplied by the Leontief inverse (*

**y***−*

**I***)*

**A**^{−1}, we obtain total output

*.*

**x***is the identity matrix so that the following matrix equation is the result of equivalence operations in our previous equation:*

**I**{\displaystyle {\vec {x}}=\left(I-A\right)^{-1}\times {\vec {y}}}

The Leontief inverse contains the multipliers for the direct and indirect inter-industry inputs required to provide 1 unit of output to final demand. Next to the inter-industry flows recorded in * Z*, each industry requires additional inputs (e.g. energy, materials, capital, labour) and outputs (e.g. emissions) which can be introduced into the calculation with the help of an environmental extension. This commonly takes the shape of an

`m`×

`n`matrix

*of total factor inputs or outputs: Factors are denoted in a total of*

**M***m*rows and the industries by which they are required are included along

*n*columns. Allocation of factors to the different industries in the compilation of the extension matrix requires careful review of industry statistics and national emissions inventories. In case of lacking data, expert opinions or additional modelling may be required to estimate the extension. Once completed,

*can be transformed into a direct factor requirements matrix per unit of useful output*

**M***, and the calculation is analogous to determination of the monetary direct multipliers matrix*

**F***(see first equation):*

**A**{\displaystyle F=M\times {\hat {x}}^{-1}}

Consumption-based accounting of resource use and emissions can be performed by post-multiplying the monetary input-output relation by the industry-specific factor requirements:

{\displaystyle E=F(I-A)^{-1}\times {\vec {y}}}

This formula is the core of environmentally extended input-output analysis: The final demand vector * y* can be split up into a domestic and a foreign (exports) component, which makes it possible to calculate the material inputs associated with each.

The matrix * F* integrates material flow data into input-output analysis. It allows us to allocate economy-wide material requirements to specific industries. With the help of the coefficients contained in the Leontief inverse (

*−*

**I***)*

**A**^{−1}, the material requirements can be allocated to domestic or foreign (exports) final demand. In order to consider variations in production structures across different economies or regions, national input-output tables are combined to form so-called multi-regional input-output (MRIO) models. In these models, the sum total of resources allocated to final consumption equals the sum total of resources extracted, as recorded in the material flow accounts for each of the regions.

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