# Production function

A production function in describing production theory the relationship between the inputs and the resulting outputs. Thus, a production function indicates the highest production quantity that a company can produce with the help of the combination of inputs.

## Types of production function

A production function is determined by the production process used for a good. A distinction is made between the following types:

### Substitutional

In the case of a substitutional production function, one production factor (at least within certain limits) can be replaced (substituted) by another or a combination of other production factors. Another characteristic of substitutionality is that the output quantity can be influenced by changing the input quantities of only one factor while the other factor quantities remain constant. With regard to substitutionality, a distinction can be made between total and peripheral substitutionality. Total substitution is when one factor can be completely replaced by another. The amount used for the factor can also be zero. Analytically calculable by solving the production factors and by deriving the isoquantum equation. Peripheral substitution is characterized by the fact that the exchange of production factors is only possible within certain limits.

The so-called CES production functions , which are characterized by a constant elasticity of substitution, form a subgroup . The best-known and most frequently used example of a substitutional production function in economics is the Cobb-Douglas production function .

#### Income law

This is probably the oldest production function. It is based on observations in agriculture and was formulated by Turgot as the law of decreasing soil yield . Two factor input quantities and one application quantity are assumed. His observations showed that the increase in labor input or fertilizer initially increases the production result, but above a certain factor input volume, the output volume steadily falls.

#### Cobb-Douglas

The Cobb-Douglas production function was developed in 1928 by Charles Wiggins Cobb and Paul Howard Douglas . In comparison to the production function under the law of income, there is no maximum in this production function, i.e. In other words, it is assumed that increasing the use of factors always results in a higher application rate. However, the return achieved by increasing the factor input decreases; i.e., if one z. If, for example, the amount used is doubled, the yield increases, but to less than double.

### Limitational

Here the factors are in a certain application ratio, i. In other words, the yield only increases if both factors are used more. However, this only applies if both factors are present to the same extent, i.e. i.e., if there is a factor in excess, this does not apply. In this case, increasing the other factor is sufficient to increase the application rate. This applies until the excess factor is used up. In order to achieve a further increase in the application rate, both factors must be increased again. Until then, the yield does not increase. This can be seen in the kink in the yield function. However, this production is only efficient if no factor is wasted, i.e. i.e., if the correct employment ratio is observed.

#### Linear-limitational

The production factors are in a fixed relationship to each other and in a fixed relationship to the output of a company or plant. A well-known representative of this type is the Leontief production function .

#### Non-linear limitational

A representative of non-linear limitational production functions is the Gutenberg production function .

### More types and newer approaches

Production type Surname content
B. Gutenberg production function Input and output quantities are related using the intensity of the processing resources; For a given performance intensity of the equipment, there are then limited production relationships between input and output
C. Heinen production function Production processes are broken down into elementary processes, the relationships between technical and economic performance can be clearly established
D. Pichler production function Throughput functions set up for limitation and substitutional models
E. Kloock production function Operation is broken down into sub-areas, multi-level production processes with cyclical interrelationships are shown and input-output matrices are set up
F. Küpper production function Input-output matrices are made dynamic by taking into account the duration of the production process
G Matthes production function A dynamic production model is developed using a network plan, which is then combined with Heinen's production function and the forms of adaptation

The theory of production functions was further developed in particular by including the environment as a natural factor of production.

The fuzzy separation between the variables input and output (or use and output) and the actual transformation has proven to be a disadvantage. Newer approaches to production theory separate the inventory variables input and output from the transformation variables consumption and generation. After all, the inclusion of factors in the company does not necessarily mean that they are also used in production (e.g. through shrinkage). Conversely, a produced good does not have to leave the company as an output (e.g. through scrap ).

The transformation can be well described by the engineering functions of technical consumption and technical generation, which enables the integration of engineering into the business production function.

In contrast to the economic and business production functions, the engineering functions focus not only on consumption and generation, but also on the technical setting and technical design of production systems.