In economics, the production function describes the amount of output in direct relation to the amount of input in the context of a production process. In simplified terms, it serves the purpose of determining the optimal ratio between input and output and thus optimizing production.

The production function is always determined by the type of production. There are two known types of this - the substitutional and the limitational production function.

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## Substitutional production function

The substitutional production function is always used when one of the production goods can be replaced (substituted). The respective substitutionality can also be taken into account in the calculation. A distinction is then made here between total and peripheral substitutionality.

The third variation is constant substitution in the context of CES functions.

The substitutional production function is essentially based on two methods of calculation:

**Income law** - This is the oldest production function, because it is based on the findings and observations of Turgot and was originally referred to by him as the "law of decreasing land yield". It shows that when the use of input increases, the output initially increases, but later and above a certain amount of input it steadily decreases.

**Cobb-Douglas** - The Cobb-Douglas function goes back to its namesake Charles Wiggins Cobb and Paul Howard Douglas. With it there is no defined production maximum and it is assumed that an increasing input always results in an increasing output.

However, it takes into account the fact that an increased input does not necessarily have to result in an output in the same ratio, but rather the output tends to decrease in a direct comparison.

## Limitation production function

In the limitational production function, it is assumed that an increase in all production factors is necessary for an increase in yield. This does not apply if only one of the production factors is in excess. As long as the excess factor is present, an increase in production is possible, but not an increase in yield.

In order to optimize the yield, the correct usage ratio must therefore be maintained. There are also two familiar schemes for this production function:

**Linear-limitational** - Here all production factors (input and output) are in a fixed relationship to one another.

**Non-linear limitational** - This scheme has no fixed definition and is still very critically, but also evaluated and researched in many ways within economics.

Well-known functions that are referred to as non-linear-limitational are the "Gutenberg function", Or the" Heinen production function "based on it.

## General information on production functions

All types of production functions are subject to constant change. Current issues are mainly the environment as Production factor oder die zunehmende globalization und steigende Komplexität der Finanzwirtschaft, welche die Entwicklungen von Produktionsfunktionen zunehmend anspruchsvoller machen.

Production functions, however, meet the mathematical criterion of additive separability without exception and thus only consist of simple summands (= function can be broken down more easily and the derivation is independent of other variables).