﻿ Inputâ€“output model

# Inputâ€“output model

The input-output analysis is a method of empirical economic research, which is used for economic analysis. It was mainly developed by Wassily Leontief, who he won a Nobel Prize in Economics.

Basis of input-output analysis is an input-output table. In her will, broken down by sector, the development of production and the intermediate products and production factors used therein (input side) and at the same time the use of the quantities (output side) produced shown.

## Input-output table

In the simplified representation provides an input -output table is as follows.

With the indices 1 to n different sectors are in fact (for example, agriculture, food industry, banking, etc.).

• The red marked part of the table contains the intermediate links. is in it for the deliveries of the sector 1 (for example agriculture) to the sector 2 (eg the food industry).
• The part marked violet contains the deliveries of sectors Endnachfrager, ie consumer goods (C ), investment goods (I) and exports (X). Since both the inputs as well as deliveries to final demand imported goods are included, while imports are deducted lump sum at the end of the line.
• The green highlighted part contains the value of the sectors ( so-called primary inputs), namely labor (L ), capital income (K ), depreciation (D) and Less indirect taxes. of subsidies ( S).

In the rows of the input-output table you will find the information for which the production ( the output) is used of each sector. In the columns you can see which primary products and factors of production, ie, what inputs are needed for production. The sum of all values ​​in a row shall be the sum of the values ​​in the corresponding column match.

## Production Theoretical assumptions of the analysis

The columns of an input-output table can be interpreted as a production function because it indicate which input materials and primary inputs (labor, capital in the form of eg machinery ) are needed to produce one unit of the good in question. In the input-output analysis it is assumed here that these factors of production relate to each other in a fixed input ratio, a so-called linear- limitational production function ( Leontief production function ). - Here, the production factor land is excluded.

## Satellite systems for Input-Output Table

Since the pure input-output table contains neither work nor ground, there are so-called satellite systems, which are written as additional rows below the input-output table. Here, employment figures are then (possibly separated by self and not self-employed) as well as capital stock and environmental factors (eg emissions of CO2)

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

The input-output analysis is also used as tool within the material flow management. It is used to determine operating indicators. For this purpose, consisting of a defined system (this can be a process or even a complete operation) exiting quantities ( = output) such as products, waste, waste water, emissions, etc. in relation with the incoming quantities ( = input) such as raw materials, auxiliary materials, power supply etc. set ( Fresner et al., 2009, page 65 to 70 ).

Ex corporate waste ratio [% ] = drop [ t ] / (raw [t ] adjuvants [ t]) * 100

## Matrix representation

The following matrices and vectors

Be the vector of total output x, the vector of final demand c, is the unit matrix E, the input-output matrix A. The coefficients of the input-output matrix a (i, j) indicate how much of the input x (i) is required to produce a unit of x ( j). A portion of the total output x is therefore as an input into the production of other outputs ( inputs ), another part remains as final demand c. It applies the following linear equation system:

Provided (E - A) is invertible.

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