Production theory

The production theory is a branch of economics and business administration . The most important sub-areas are the neoclassical production theory, which works with production functions , activity analysis and the theory of forms of adaptation according to Gutenberg.

In economics, it describes the derivation of the supply curve in the market model . Starting from a technology that describes all technically feasible combinations of input factors , the most efficient combination of factors - for given prices - can be derived (so-called profit maximization ). From this, factor demand and the supply of goods can be derived.

From the point of view of business administration, the aim of production theory is to use production functions to show relationships between the quantitative use of factors and the resulting output volume . The production theory is supplemented by the cost theory , which deals with the functional relationships between the costs that arise from the use of factors and the output achieved or the level of output. The combination of production factors can be assessed according to their technical and economic efficiency (e.g. economies of scale , composite effects ).

General

Production functions

A production function creates a connection between input and output. In the general case it is a function of form

${\ displaystyle f (x_ {1}, x_ {2}, \ dotsc, x_ {n}, r_ {1}, r_ {2}, \ dotsc, r_ {m}) = 0}$ . This representation is called the production equation.

The representation is called the product function. In the case of a single product, it is simplified to ${\ displaystyle (x_ {1}, x_ {2}, \ dotsc, x_ {n}) = f (r_ {1}, r_ {2}, \ dotsc, r_ {m})}$ ${\ displaystyle x = f (r_ {1}, r_ {2}, \ dotsc, r_ {m})}$

The representation is called the factor function. In the case of a single factor, it simplifies to ${\ displaystyle (r_ {1}, r_ {2}, \ dotsc, r_ {m}) = f (x_ {1}, x_ {2}, \ dotsc, x_ {n})}$ ${\ displaystyle r = f (x_ {1}, x_ {2}, \ dotsc, x_ {n})}$

For particularly common types of production functions, see Production Function .

The quotient is called the production coefficient. In the case of linear functions it is constant. ${\ displaystyle a_ {i} = {\ frac {r_ {i}} {x}}}$

Substitutionality and limitationality

A production function is limitational if, for a given production quantity, there is only one single combination of factors with which it can be realized. This means that factors cannot be exchanged with one another. The function is substitutional if there are several possible combinations of factors for a given production quantity. One distinguishes between:

partial substitution, in which factors cannot be completely exchanged. Example is the function . ${\ displaystyle x = r_ {1} \ cdot r_ {2}}$
total substitution in which one or more factors can be completely replaced. Example is . ${\ displaystyle x = r_ {1} + r_ {2}}$

Factor variation

With partial factor variation, at least one factor is variable, but not all. It is considered in which direction and how much the output changes or with which factors it changes at all.

Partial marginal yield ${\ displaystyle PG_ {i} = {\ frac {\ partial x} {\ partial r_ {i}}}}$

In the total analysis, all factors are variable.

Total marginal product ${\ displaystyle \ Delta x = {\ frac {\ partial x} {\ partial r_ {1}}} \ Delta r_ {1} + \ dotsb + {\ frac {\ partial x} {\ partial r_ {n}} } \ Delta r_ {n}}$

Level variation ${\ displaystyle x = f (t \ cdot r_ {1}, t \ cdot r_ {2}, \ dotsc, t \ cdot r_ {m})}$

Scale elasticity

${\ displaystyle \ eta = {\ frac {t \ cdot dx} {x \ cdot dt}}}$

${\ displaystyle \ eta <1}$ decreasing returns to scale
${\ displaystyle \ eta = 1}$ constant returns to scale
${\ displaystyle \ eta> 1}$ increasing returns to scale

homogeneity

One calls f homogeneous of degree if holds ${\ displaystyle \ epsilon}$ ${\ displaystyle t ^ {\ epsilon} \ cdot x = f (t \ cdot r_ {1}, t \ cdot r_ {2}, \ dotsc, t \ cdot r_ {m})}$

${\ displaystyle \ epsilon <1}$ disproportionately
${\ displaystyle \ epsilon = 1}$ linear
${\ displaystyle \ epsilon> 1}$ disproportionately

Ways of looking at things

In the long-term view, it is assumed that all production factors r are variable, in the short-term some factors are fixed.

Models

Various models exist in production theory. The oldest are based on production functions that establish a direct connection between the quantities of factors used and the quantities of product produced in the process, without justifying this technologically. The activity analysis assumes a number of technically feasible production possibilities (referred to there as technology) and analyzes them. The Engineering Production Functions assume that there are many technical options during planning and consider these for many different special cases. The Gutenberg production function and the functions based on it, on the other hand, are based on an already existing production system and explicitly differentiate between the factors of operating resources that can be used over and over and materials that are consumed. The putty-clay model combines both approaches: during the planning of production systems, you have many options here as with the engineering production functions, but during operation hardly as with the activity analysis and the Gutenberg production function.

Technical efficiency

When examining technical efficiency, it is not just the final sum of the factors of production that counts, but rather the possible combinations and alternatives. The following example should make this clear:

To produce a product x, two factors f1 and f2 are required. The application rate is 4 units each. The following combinations of production factors are technically possible:

possibility	Factor f1 (ME)	Factor f2 (ME)	Sum f1 + f2 (ME)	Output x (ME)
a	1	5	6th	4th
b	2	3	5	4th
c	2	5	7th	4th
d	3	2	5	4th
e	3	3	6th	4th
f	3	7th	10	4th
G	4th	2	6th	4th
H	5	1	6th	4th
i	6th	1	7th	4th

The combinations b and d are obviously technically efficient, since fewer production factors are required in total.

Option a is also technically efficient because, compared to option c, it also requires 5 units of the factor f2, but only one unit of the 1st factor. The same applies to the combination h. Compared to the combination i, it manages with 5 units f1 when using one unit f2.

Although the combination g only requires 6 units in total, it is not technically efficient because the more efficient combination d exists for the use of 2 units f2.

If you compare the quantitative change of two combinations, you get the marginal rate of technical substitution , which results from the ratio of the quantitative change of the replaced factor to that of the replacing factor.

Example for changing the combinations from a to b: 2 ME of factor f2 are replaced by 1 ME of factor f1. The dividend is therefore 2/1 = 2 (positive slope). If you switch from combination d to h, the limit rate is 1/2 = 0.5 (negative slope).

Economic efficiency

The minimum cost combination results from the economic consideration . If you determine the factor prices for the above example as follows, the following costs result:

possibility	Factor f1 (ME)	Value f1 (30 GE)	Factor f2 (ME)	Value f2 (20 GE)	Sum f1 + f2 (GE)	Output x (ME)
a	1	30th	5	100	130	4th
b	2	60	3	60	120	4th
c	2	60	5	100	160	4th
d	3	90	2	40	130	4th
e	3	90	3	60	150	4th
f	3	90	7th	140	230	4th
G	4th	120	2	40	160	4th
H	5	150	1	20th	170	4th
i	6th	180	1	20th	200	4th

The combination b ensures the lowest costs of 120 goods units (GE).

literature

Dyckhoff, Spengler: Production Management: An Introduction 3rd Edition, Springer, Heidelberg, 2010.

Individual evidence

↑ Wöhe: Introduction to Business Administration . 19th edition. Verlag Franz Vahlen, Munich 1996, p. 476 ff.

↑ Busse von Colbe: Business Theory: Volume 1 Basics, Production and Cost Theory . 5th edition. Springer, p. 102

↑ Busse von Colbe: Business Theory: Volume 1 Basics, Production and Cost Theory . 5th edition. Springer, p. 101

↑ Busse von Colbe: Business Theory: Volume 1 Basics, Production and Cost Theory . 5th edition. Springer, pp. 105f.

↑ Busse von Colbe: Business Theory: Volume 1 Basics, Production and Cost Theory . 5th edition. Springer, p. 110.

^ Corsten: Production Management 12th Edition, p. 52.

^ Corsten: Production Management 12th Edition, p. 53.

^ Corsten: Production Management 12th Edition, p. 54.

↑ Busse von Colbe: Business Theory: Volume 1 Basics, Production and Cost Theory . 5th edition. Springer, p. 101

[1] Wöhe: Introduction to Business Administration . 19th edition. Verlag Franz Vahlen, Munich 1996, p. 476 ff.