Gutenberg production function

The U-shaped course of the production coefficient a as a function of the intensity d is typical of the Gutenberg production function

The Gutenberg production function (also: production function type B or theory of forms of adaptation ) is a production function developed by Erich Gutenberg . It emerged from the production function under the law of income , which is also known as type A production function and was adopted from economics in business administration . The Gutenberg production function, together with activity analysis, is one of the two most important innovations in production and cost theory in the second half of the 20th century.

The Gutenberg production function, like all other production functions, is a theoretical model that considers relationships between the quantities of raw materials used in production and the quantities produced in the process. It is the first production function that is geared towards the special requirements of business administration and forms the basis of many other production functions (types C , D , E and F ). While earlier production functions indicate the maximum possible production amount x for a given set of production factors r (workers, machines, etc.), in Gutenberg both the production amount and the factor consumption result from the number n of active machines, the speed d with which they work (intensity), and the time t during which they are active.

history

Beginnings in economics

The production theory emerged in the 18th century as a branch of price theory , which is itself part of the political economy is thus even before the development of business administration in 1900. Using the production features that for a given amount of production factors labor and capital indicates the maximum possible amount of products , an attempt is made to derive the supply curve of the companies. It shows how much the companies are offering for a given price. Along with the demand curve of the households can be a then market equilibrium determined. The production functions used in economics assumed that a certain amount of production can be achieved through various combinations of labor and capital, which stems from the observation that in the course of the industrial revolution human labor was replaced by machines - which are part of real capital. The property that the factors of production are interchangeable is called substitutionality. In addition, the early production functions had the property that even if one factor was constant, the production quantity could be increased by increasing the use of the other factor. Based on the production function, the consumption of the factors was evaluated with the respective procurement prices in order to arrive at the cost functions . With them it was then possible to derive the quantity offered by the companies for given market prices of the products.

The production function under the law of income , which was developed on the basis of observations in agriculture at the end of the 18th century, was the first concrete production function. On the basis of conclusions by analogy , the prevailing opinion was that it was also valid for industrial production. It was further developed into the Cobb-Douglas function in the 19th century . Both are substitutional production functions. The entire theory of production was integrated into the business administration theory that emerged around 1900. The connection between production and the costs caused by it was of particular interest within industrial management .

Gutenberg's criticism of the state of production theory

Gutenberg was familiar with these theories through his studies in economics. After completing his studies, he worked for an industrial company for several years, then studied business administration and finally became a professor of business administration. In the second edition of his major work "Die Produktion" from 1955, he criticized the previous theory in general and, above all, the income law in particular as unsuitable for industrial practice, although he defended it in the first edition from 1951. The classical theory implicitly assumes that the factors labor and capital can be divided at will. However, workers and machines are not freely divisible. Substitutionality was also criticized: it is not fulfilled in operational reality. Instead, so-called limitationale conditions prevail in which only a certain combination of factors leads to the desired production quantity. For example, a certain number of operators is required for a certain type of machine. More staff on a single machine cannot increase the production volume. The number of machines then limits the maximum possible production volume. In general, the previous theory abstracts from how the factors are transformed into products and therefore leads to the fact that it has not proven itself very well.

The new production function

Gutenberg developed a new production function, which he called “type B production function”, as a counterpart to the income law, which he called “type A production function”, which has since been known under this name in business administration. At first he divided the production factors into human labor, machines and materials and placed machines at the center of his consideration. He saw the number of machines as constant; however, the production volume is variable based on the number of machines used (quantitative adjustment) , the intensity with which they work (in pieces produced per time, intensity-related adjustment ) and through their duration of use (temporal adjustment) . He assumed that doubling the number of machines (with constant intensity and duration of use) would also double the consumption of materials. The same applies to a change in the duration of the assignment. If the intensity is changed, however, no general statement can be made about the consumption of the other factors. These must be determined separately for each machine and each factor. On many machines, the consumption of operating materials (fuel, lubricating oil) per manufactured product follows a U-shaped curve with a consumption-minimal intensity. This case was examined in detail by Gutenberg: For every possible production volume, he determined the combination of number of machines, duration of use and intensity that leads to the lowest costs.

reception

Gutenberg's work met with great approval. His theories determined the direction of research in business administration for about a decade. The science program he founded is called the factor theoretical approach . The division of production factors proposed by him was refined and expanded, but the basic structure was retained. There was also much work on its production function that supplemented and confirmed the theory. Edmund Heinen developed it further into the Heinen production function named after him . However, he was not satisfied with the state of the theory: “Business economies are not 'events' for any abstract production factors, but social systems in which people […] work together.” The science program he initiated determined business research in the following decade and is more decision-oriented Approach . The Gutenberg production function, together with the Income Act, has retained a permanent place in business textbooks to this day. While the latter is listed more for didactic reasons, the former is certified as being highly practical.

The z situation

The technical parameters of the individual machines are summarized in the so-called z-situation, which is then viewed as constant. It includes all values ​​that are relevant for production but cannot be changed in the short term. An engine can be characterized, for example, by the number of its valves (z ₁ ), the displacement (z ₂ ), the type of fuel required (z ₃ ), the operating temperature (z ₄ ) and other features that are determined by the design. If machines can be converted (for example, punching machines can be equipped with differently shaped tools), this means a transition to a different z-situation. The stock of machines is also included; The acquisition of new or the sale of existing machines is therefore assessed as a change in the z-situation. Of all the technical values, the intensity is  particularly emphasized. It indicates the economic units of work performed per time by a machine, e.g. B. Pieces per hour (for piece goods) or meters per minute (for fluid goods). ${\displaystyle d}$

The consumption function

Earlier production functions presented a direct relationship between in production volume (also: output volume or output) and the amount of consumed factors (mostly R ohstoffe, as amount used, or input referred). These functions were in the form of a single factor or for various factors. In the Gutenberg production function, both the production volume and the factor consumption depend on the intensity of production , the duration of production and the number of active machines . ${\displaystyle x}$ ${\displaystyle r}$ ${\displaystyle x=f(r)}$ ${\displaystyle x=f(r_{1},r_{2},\cdots ,r_{n})}$ ${\displaystyle n}$ ${\displaystyle d}$ ${\displaystyle t}$ ${\displaystyle m}$

${\displaystyle x=d\cdot t\cdot m}$

${\displaystyle {\begin{aligned}r&=a\cdot x\\&=a\cdot d\cdot t\cdot m\end{aligned}}}$

It is the so-called production coefficient , which generally is not constant but depends on the intensity, so . With this form of representation, it was assumed that all machines are identical and are operated with the same intensity and duration. ${\displaystyle a}$ ${\displaystyle a(d)}$

If you want to take into account different intensities and usage times, you have to add the production quantities of the individual machines for the total production quantity and the consumption of the individual machines for the total factor consumption:

${\displaystyle x=d_{1}\cdot t_{1}+d_{2}\cdot t_{2}+\dotsb +d_{m}\cdot t_{m}}$

${\displaystyle r=a_{1}\cdot x_{1}+a_{2}\cdot x_{2}+\dotsb +a_{m}\cdot x_{m}}$

As a rule, several factors are required at the same time for production (e.g. tools, fuels, raw materials, lubricants, etc.). A separate consumption function must therefore be determined for all factors. ${\displaystyle n}$

The production coefficient can be interpreted as average consumption (consumption per piece produced). For many types of factors there is a U-shaped curve with a minimal average consumption of intensity . The associated consumption functions then have an S-shaped curve. This case was examined in detail by Gutenberg. ${\displaystyle \textstyle a(d)={\frac {r}{x}}}$ ${\displaystyle d_{\mathrm {opt} }}$ ${\displaystyle r}$

Cost functions of pure adjustments

→ Main article : Adaptation (business administration)

If the unit costs remain constant (blue), the result is a linearly increasing total cost function (red)

Cost functions establish a connection between the production volume and the costs arising from production . The total costs of an individual machine are calculated by multiplying the consumption of the factors by the respective procurement prices and then adding the costs of the individual factors: ${\displaystyle x}$ ${\displaystyle K}$ ${\displaystyle p}$

${\displaystyle {\begin{aligned}K&=r_{1}p_{1}+r_{2}p_{2}+\cdots +r_{n}p_{n}\\&=a_{1}(d)x_{1}p_{1}+a_{2}(d)x_{2}p_{2}+\cdots +a_{n}(d)x_{n}p_{n}\end{aligned}}}$

If you divide the total costs by the production volume, you get the unit costs : ${\displaystyle k}$

${\displaystyle k=K/x=a_{1}(d)p_{1}+a_{2}(d)p_{2}+\cdots +a_{n}(d)p_{n}}$

Fixed costs that arise regardless of whether or how much is produced are usually neglected because production has no influence on them. Strictly speaking, it is therefore a question of the variable costs and the variable unit costs . ${\displaystyle K_{v}}$ ${\displaystyle k_{v}}$

Adjustment in terms of intensity

Constant total costs and falling unit costs

Unit costs rising linearly and total costs rising over-linearly

u-shaped course of the unit costs and s-shaped course of the total costs

With the purely intensity-based adjustment, the number of machines and the duration of use are kept constant and only the intensity of the machines is changed between the minimum possible intensity and the maximum possible intensity . They are often given by technology: for example, many motors only work within a certain speed range. The minimum intensity can, however, be zero. In the following, for the sake of simplicity, it is assumed that production is carried out with a single machine. The course of the cost functions depends on the course of the production coefficients . ${\displaystyle d_{\mathrm {min} }}$ ${\displaystyle d_{\mathrm {max} }}$ ${\displaystyle a(d)}$

With production coefficients that are constant, i.e. independent of the intensity, there is a horizontal course of the unit costs and a linearly increasing course of the total costs. This special case was examined by Wassily Leontief in the Leontief production function named after him before Gutenberg . This is often the case for the materials that make up the end product. The production coefficient often results directly from parts lists , recipes, chemical reaction equations and the like. For example, exactly two tires are required to manufacture a bicycle, regardless of how quickly the production takes place. If the workers are paid by piece wages, i.e. per manufactured product, then there is also a constant production coefficient for the “consumption” of the workers.

Falling production coefficients result if workers are paid per time ( time wage ). The faster you work, the lower the unit costs. The total costs then run horizontally, i.e. regardless of how quickly the production takes place.

Increasing production coefficients are typical for the consumption of lubricants or tools in turning or milling machines . Consumption is often analyzed in more detail in engineering . The wear of many tools is shown, for example, by the Taylor straight line . The consumption of materials also increases above a certain threshold when the increased production speed leads to increased rejects .

A U-shaped course has proven to be typical for many operating materials. The production coefficient then first drops until it reaches its minimum and then rises again. The course of the unit costs is then also U-shaped and the associated total costs are S-shaped: They rise quickly at first, then more and more slowly until they rise approximately linearly in the vicinity of . Then they rise faster and faster. If you add the unit costs of the individual factors on a single machine, the result is often a U-shaped curve. If, for example, a machine only needs the two factors work (time wages, i.e. falling unit costs) and tools (increasing), the total unit costs are U-shaped. For the combined adjustments (intensity-quantitative, intensity-temporal and intensity-temporal-quantitative), a U-shaped production coefficient is therefore always used below . ${\displaystyle d_{\mathrm {opt} }}$ ${\displaystyle d_{\mathrm {opt} }}$ ${\displaystyle a(d)}$

Purely temporal adjustment

With purely temporal adjustment, the number of machines and their intensity are constant; only their duration of use is changed. In this case, the production coefficients are constant and there are constant unit costs and linearly increasing total costs - just like with the intensity-based adjustment and constant production coefficient. The time can only be varied within the limits and . The minimum duration of production is usually zero, but it can also be the same as the maximum duration. Starting up power plants and blast furnaces , for example, is so expensive that they are normally in continuous use and only switched off for maintenance work. The maximum working time can be specified technically, but is usually determined by the contractually or collectively regulated working hours. Adjusting the time is usually not a problem for the machines: They are simply switched off after the desired duration. However, workers who are paid according to time wages must be paid even if they are not working. A certain amount of adaptability arises through the use of short-time work and the reduction of shifts or through overtime and additional shifts. Flexible working time models are also possible . ${\displaystyle t_{\mathrm {min} }}$ ${\displaystyle t_{\mathrm {max} }}$

Purely quantitative adjustment

With the purely quantitative adjustment, the intensity and duration of use of all active machines are identical. Previously inactive machines are put into operation or active ones are switched off. The "course" of the cost functions consists of individual points. If the production quantity can be generated with a single machine , the cost functions are only defined at the points . However, a distinction must be made as to whether the existing machines all have the same unit costs (mutative adaptation) or different (selective adaptation) . In many companies, new machines are procured over and over again over many years, so that, in addition to newer and generally low-consumption machines, there are also older ones with higher consumption. The selective adjustment is therefore the normal case. As a rule, however, it is easily possible to adjust the time of all machines at the same time, so that the purely quantitative adjustment is rarely analyzed in detail. It is viewed as a special case of combined temporal and quantitative adjustment. ${\displaystyle x}$ ${\displaystyle x,2x,3x,\dotsc }$

Cost functions of combined adjustments

In the combined adjustments are at least two or all three possible parameters and changed simultaneously. It is therefore a combination of the pure adjustments. ${\displaystyle d,t}$ ${\displaystyle m}$

Temporal and quantitative adjustment

Total cost trend with time-quantitative adjustment

With the combined temporal and quantitative adjustment, it is usually assumed that fixed costs are incurred in a company that are independent of the production volume, such as rents for buildings, insurance fees or leasing fees for machines. In addition, so-called jump fixes or fixed interval costs are incurred for the mere commissioning of each individual machine . ${\displaystyle K_{\mathrm {fix} }}$ ${\displaystyle K_{fi}}$

With identical machines (mutative adjustment) , all sections of the cost functions have the same slope. The fixed costs can be broken down into a utility cost component and an idle cost component, which depends on the production volume. If the maximum possible production quantity is designated with and the actual quantity with , the idle costs result as: ${\displaystyle x_{\mathrm {max} }}$ ${\displaystyle x}$

${\displaystyle K_{\mathrm {leer} }={\frac {K_{\mathrm {fix} }}{x_{\mathrm {max} }}}\cdot (x_{\mathrm {max} }-x)}$

and utility costs as

${\displaystyle K_{\mathrm {nutz} }={\frac {K_{\mathrm {fix} }}{x_{\mathrm {max} }}}\cdot x}$

In the case of machines with different cost profiles, there is an economic selection problem (selective adaptation) . Small production quantities are produced with the more cost-effective machine. Only when their capacity is no longer sufficient is an additional machine used for production. If the less expensive machine is shut down first when the production volume declines, the costs remain at a higher level than was the case before the expansion of production. This effect is known as cost retention. If, on the other hand, capacities are consciously built up in order to be able to expand the production volume in the future, these additional capacities still cause costs. This phenomenon is called cost precision .

Adjustment in terms of intensity and time

Unit and total costs with time-intensity adjustments in the Gutenberg production function

The combined intensity and time adjustment is usually considered for the case of a single machine and U-shaped production coefficient. In the following, for the sake of simplicity, it is assumed that the minimum duration of use and the minimum intensity are both zero. Because of the U-shaped production coefficient, there is a consumption-minimum intensity at which production is preferred. Small production quantities are then always produced and the duration of use is adapted to the required production quantity. However, a maximum of product units can be produced with this method . If more is to be produced, the intensity must be increased for the maximum duration of use . Overall, it can then be stated that production quantities that are smaller than are achieved through purely temporal adjustment at and , and those that are larger through purely intensity-related adjustment at and . ${\displaystyle t_{\mathrm {min} }}$ ${\displaystyle d_{\mathrm {min} }}$ ${\displaystyle d_{\mathrm {opt} }}$ ${\displaystyle d_{\mathrm {opt} }}$ ${\displaystyle x_{\mathrm {opt} }=d_{\mathrm {opt} }\cdot t_{\mathrm {max} }}$ ${\displaystyle d_{\mathrm {opt} }}$ ${\displaystyle x_{\mathrm {opt} }}$ ${\displaystyle d=d_{\mathrm {opt} }}$ ${\displaystyle t\leq t_{\mathrm {max} }}$ ${\displaystyle t=t_{\mathrm {max} }}$ ${\displaystyle d\geq d_{\mathrm {opt} }}$

Graphically, this procedure means a combination of the pure forms of adaptation for the unit cost trend. It is initially horizontal like the unit cost curve with purely temporal adjustment and from then on it increases like the curve with purely intensity-based adjustment. The total costs accordingly initially increase linearly from the origin to increase more and more. ${\displaystyle x=x_{\mathrm {opt} }}$ ${\displaystyle x_{\mathrm {opt} }}$

Adjustment in terms of intensity, quantity and time

Course of the marginal costs of two machines with, temporal, quantitative, intensity-based adjustment.

In the case of the combined intensity, quantitative and temporal adjustment, it is mostly assumed that the cost functions for the combined intensity and temporal adjustment are already known for the individual machines. The total costs of the individual machines are then derived from the production volume in order to arrive at the marginal costs ${\displaystyle K'}$

${\displaystyle K'={\frac {\mathrm {d} K(x)}{\mathrm {d} x}}}$.

Geometrically, the marginal costs can be interpreted as the slope of the cost function. From an economic point of view, they indicate for a certain production quantity by how much the costs increase if an additional ( infinitesimally small) unit is produced. The marginal costs are basically similar to the unit costs with time and intensity adjustments. ${\displaystyle x}$ ${\displaystyle K}$

Basically, the production volume is expanded on the machine that has the lowest marginal costs. It will first be adjusted in time. Since the marginal costs do not increase during the time adjustment due to the constant production coefficient, the duration of use is increased until the maximum duration of use is reached. After that, the intensity has to be adjusted, which leads to rising marginal costs. As soon as you have reached the marginal costs of the second cheapest machine, this is adjusted in time.

Critical appreciation and further development

The Gutenberg production function is said to be very realistic, especially for the industrial production of piece goods (manufacturing industry), and to a lesser extent for the chemical and raw materials industry (process industry). However, it does not fully meet the operational possibilities. It does not take into account the temporal course of production, but only the duration of use and also no order sequences , lot sizes and substitutional production conditions. The latter are particularly prevalent in the chemical industry. There, different combinations of temperature and pressure often lead to the same result. There are consequently options for the output of the compressor (for the pressure) and the oven (for the temperature). Edmund Heinen has taken up the mentioned points of criticism and expanded and refined the Gutenberg production function to the Heinen production function , which is also known as the "type C production function". It is also criticized that the Gutenberg production function does not adequately reflect the use of potential factors (system wear and human work). The z-situation, viewed as constant, is also criticized: every time it changes, new consumption functions have to be determined. Overall, it is therefore particularly suitable for modeling material consumption (energy, fuel, tools, lubricants, etc.) in the manufacturing industry.

In the 1960s several studies dealt with empirical verification of the Gutenberg production function. In 1960 the relationship between the energy generated and the consumption of steam in a steam turbine was determined. In 1966 the energy consumption of electric motors and the coal consumption of steam boilers were examined. In 1968, the consumption of water and steam was determined on a paper production plant. In all of them, a production coefficient was determined that was dependent on the intensity and mostly also had a U-shaped curve. Only the consumption of the paper plant decreased monotonously.

Inspired by Karl Popper's work on critical rationalism , Marcel Schweitzer and Hans-Ulrich Küpper examined a large part of the production theory of the time in 1974 . The Gutenberg production function turned out to be free of contradictions and basically universally valid - at least within the industrial production for which it was developed. In principle, it can also be checked empirically, even if this can be difficult in detail due to measurement problems. It was certified to have a significantly higher degree of proof than other production functions and a relatively large scope. Only the lack of separation into axioms and the logical conclusions derived from them (“ axiomatization ”) was criticized.

Parallel developments in production theory

At the same time as Gutenberg's work, there were two further developments in production theory in the Anglo-Saxon field:

• In economics, linear activity analysis was developed. It was generalized at the end of the 1960s by Waldemar Wittmann and, like all economic developments before, it was incorporated into German business administration. The general activity analysis then contains the Gutenberg production function as a special case, as well as all other production functions.
• The Engineering Production Functions were created in Operations Management, the Anglo-Saxon counterpart to production management . Like the Gutenberg function, they are characterized by a high degree of practical orientation, but relate to production facilities that are currently being planned and not to the optimization of existing companies.

See also

• Production engineering
• Cost and performance accounting
• Work preparation

literature

Sunday: The Gutenberg production function , Gabler, Wiesbaden, 2004

Individual evidence

1. ↑ Wöhe: Introduction to general business administration, 25th edition, p. 310.
2. ^ Fandel: Production I , 1st edition 1987, Springer, Berlin, p. I
3. ^ Dyckhoff: Betriebliche Produktion Springer, Berlin, 1992, p. 35
4. ↑ Steven: Production Theory Gabler, Wiesbaden, 1998, p. 21f.
5. ↑ Steven: Production Theory Gabler, Wiesbaden, 1998, p. 34f.
6. ↑ Sunday: Die Gutenberg production function , Gabler, Wiesbaden, 2004, p. 11
7. ↑ Heinen: Basic questions of decision-oriented business administration In: Schweitzer (Hrsg.): Opinions and scientific goals of business administration. Darmstadt 1978, pp. 219-246. Quoted from: Erich Zahn, Uwe Schmidt: Production Management. Volume 1: Basics and operational production management. UTB, Stuttgart, ISBN 978-3-8252-8126-7 , p. 25. Full quote: “Business economies are not 'events' of any abstract production factors, but social systems in which people use technical aids to share work and cooperate to achieve the organizational goal and work together with your own goals. "
8. ↑ Steven: Production Theory , Gabler, Wiesbaden, 1998, p. 128.
9. ^ Steven: Production Theory Gabler, Wiesbaden, 1998, pp. 130 + 133.
10. ^ Corsten: Production Management 12th Edition, pp. 145f.
11. ↑ Steven: Production Theory Gabler, Wiesbaden, 1998, pp. 145–150.
12. ↑ Steven: Production Theory Gabler, Wiesbaden, 1998, p. 145.
13. ↑ Steven: Production Theory Gabler, Wiesbaden, 1998, pp. 145–150.
14. ↑ Steven: Production Theory Gabler, Wiesbaden, 1998, pp. 135f.
15. ^ Corsten: Production Management 12th Edition, p. 150f.
16. ↑ Steven: Production Theory Gabler, Wiesbaden, 1998, pp. 138–143.
17. ^ Corsten: Production Management 12th Edition, p. 150f.
18. ^ Corsten: Production Management 12th Edition, p. 152
19. ^ Corsten: Production Management 12th Edition, p. 153f.
20. ^ Adam: Production Management 9th Edition, p. 380
21. ^ Corsten: Production Management 12th Edition, p. 101
22. ↑ Gälweiler: production costs and production rate in 1960
23. ↑ Pack determination of the minimum cost adjustment process combination, in: Zeitschrift für Betriebswirtschaftliche Forschung 1966, pp. 466–476
24. ^ Pressmar The cost-performance function of industrial production plants 1968
25. ^ Fandel: Production I 5th edition, pp. 204–216
26. ↑ Schweitzer M., Küpper H.-U .: Production and cost theory of the Reinbek company near Hamburg, 1974
27. ^ Fandel: Production I 5th edition, pp. 195–198 and 201