Input-output analysis

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The input-output analysis (also input-output analysis , or input-output decomposition ) is a method of empirical economic research that is used for economic analyzes. It was mainly developed by Wassily Leontief , who received the Nobel Prize in Economics for it .

The input-output analysis is based on an input-output table . It shows the development of production and the intermediate products and production factors used (input side) and, at the same time, the use of the quantities produced (output side), broken down according to economic sectors .

Input-output table

In the simplified representation, an input-output table looks like this.

${\displaystyle a_{1,1}}$	${\displaystyle a_{1,2}}$	...	${\displaystyle a_{1,n}}$	${\displaystyle C_{1}}$	${\displaystyle I_{1}}$	${\displaystyle X_{1}}$	${\displaystyle -M_{1}}$
${\displaystyle a_{2,1}}$	${\displaystyle a_{2,2}}$	...	${\displaystyle a_{2,n}}$	${\displaystyle C_{2}}$	${\displaystyle I_{2}}$	${\displaystyle X_{2}}$	${\displaystyle -M_{2}}$
...	...		...	...	...	...	...
${\displaystyle a_{n,1}}$	${\displaystyle a_{n,2}}$	...	${\displaystyle a_{n,n}}$	${\displaystyle C_{n}}$	${\displaystyle I_{n}}$	${\displaystyle X_{n}}$	${\displaystyle -M_{n}}$
${\displaystyle L_{1}}$	${\displaystyle L_{2}}$	...	${\displaystyle L_{n}}$
${\displaystyle K_{1}}$	${\displaystyle K_{2}}$	...	${\displaystyle K_{n}}$
${\displaystyle S_{1}}$	${\displaystyle S_{2}}$	...	${\displaystyle S_{n}}$
${\displaystyle D_{1}}$	${\displaystyle D_{2}}$	...	${\displaystyle D_{n}}$

Various sectors are identified with the indices 1 to n (e.g. agriculture, food industry, banking, etc.).

• The part of the table marked in red contains the wholesale activities . stands for the supplies of Sector 1 (e.g. agriculture) to Sector 2 (e.g. the food industry). ${\displaystyle a_{1,2}}$
• The part marked in purple contains the deliveries of the sectors to end users , i.e. consumer goods (C), capital goods (I) and export goods (X). Since both intermediate consumption and deliveries to final demand contain imported goods, imports are deducted at the end of the line.
• The green part contains the added value of the sectors (so-called primary inputs ), namely labor (L), capital income (K), depreciation (D) and indirect taxes minus. of subsidies (S).

In the lines of the input-output table you can find information about what the production ( output ) of each sector is used for. In the columns you can see which preliminary products and production factors, i.e. which inputs, are required for production. The sum of all values ​​in a row must match the sum of the values ​​in the corresponding column.

Production-theoretical assumptions of the analysis

The columns of an input-output table can be interpreted as a production function , since they indicate which input materials and primary inputs (labor, capital in the form of e.g. machines) are required to produce a unit of the relevant good. In the input-output analysis, it is assumed that these production factors are in a fixed relationship to one another, a so-called linear-limitationale production function ( Leontief production function ). - The production factor soil is excluded.

Satellite systems for input-output table

Since the pure input-output table contains neither work nor land , there are so-called satellite systems that are written as additional lines below the input-output table. Here you can find employment figures (if necessary, separated into self-employed and not self-employed) as well as capital stock and ecological factors (e.g. emissions of CO ₂ )

The input-output analysis as an instrument of material flow management

The input-output analysis is also used as an instrument within material flow management. It is used to determine operational key figures. For this purpose, the quantities of substances (= output) emerging from a defined system (this can be a process or a complete operation) such as products, waste, wastewater, emissions, etc. are compared with the quantities of substances (= input) such as raw materials, auxiliary materials , Energy supply etc. set.

Example: Company waste rate [%] = waste [t] / (raw materials [t] + auxiliary materials [t]) * 100

Matrix display

The following matrices and vectors

${\displaystyle x={\begin{pmatrix}x_{1}\\\vdots \\x_{4}\end{pmatrix}};\qquad c={\begin{pmatrix}c_{1}\\\vdots \\c_{4}\end{pmatrix}};}$

${\displaystyle E={\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}};\qquad A={\begin{pmatrix}a_{11}&\cdots &a_{14}\\\vdots &\ddots &\vdots \\a_{41}&\cdots &a_{44}\end{pmatrix}};}$

be the vector of the total output x , the vector of the final demand c , the unit matrix E , the input-output matrix A . The coefficients of the input-output matrix a (i, j) indicate how much of input x (i) is needed to produce a unit of x (j) . A part of the total output x is used as input in the production of other outputs (intermediate consumption), another part remains as final demand c . The following system of linear equations applies :

${\displaystyle {\begin{matrix}Ax+c&=&x\\(E-A)x&=&c\\x&=&(E-A)^{-1}c\end{matrix}}}$

provided (I - O) can be inverted .

${\displaystyle {\begin{matrix}Ax\end{matrix}}}$specifies what of the total production x is required as input for the production of x itself.

See also

• Energy and material flow management

literature

• Johannes Fresner, Thomas Bürki, Henning H. Sittel: Resource efficiency in production - reducing costs with cleaner production. Symposion Publishing, Düsseldorf 2009, ISBN 978-3-939707-48-6 .
• Federal Statistical Office, Series 18 Series 2, National Accounts, Input-Output Calculation. Wiesbaden 2006.

Individual evidence

1. ↑ Fresner et al., 2009, pp. 65 to 70.