**3 Mathematical description of an employment structure**

**Primary parameters:**

- There are
**a**workers. - These workers are employed in
**m3**different professions, which are numbered with the index**j3**. - It is assumed that the performance within a profession can be measured. Accordingly there can be a special unit of measurement or a
**normal Md**for each profession._{j3}for work performance - A worker
**j1**has a certain individual skill for each profession j3 to achieve a corresponding productivity**P**with normal effort. The totality of all productivities P_{j1,j3}_{j1,j3}of a worker j1 results in a vector**P**, which we call productivity profile. Usually every worker has its own individual productivity profile. To reduce the math we assume that there can be several workers with the same productivity profile. These workers are combined to form one productivity group._{j1} - Accordingly, the assumption is: There are
**m2**different**productivity groups**, which are numbered with the index**j2**. The productivity profile of group j2 is given by the productivity profile vector P_{j2}, which consists of the components P_{j2,j3}. The vectors P_{j2}of the productivity groups can be combined into a matrix, which we call the matrix of productivity profiles or briefly the productivity matrix P of society. - The components P
_{j2,j3}of the productivity matrix P are made dimensionless with the corresponding normal_{j3}, so that there is the**dimensionless productivity matrix p**with the dimensionless components**p**._{j2,j3}

Each productivity group j2 includes

**ap**workers who have the corresponding productivity profile. All these numbers can be summarized in a vector_{j2}**ap**, which consists of the m2 components ap_{j2}and which we call the**vector of the productivity group sizes**.

In fact, every worker has its own productivity profile. By introducing the productivity groups we have not made any significant restriction to the general case because if we set all productivity group sizes equal to one then again an individual productivity profile is given for each worker.

- When assuming full employment the sum of all productivity group sizes is equal to the number a of workers.

- The workers in a productivity group can be employed in different professions. For example, in profession j3 from productivity group j2 the number of
**ac**workers can be employed. This statement can be made for any productivity group and any profession. All numbers ac_{j2,j3}_{j2,j3}for j2=1 to m2 and j3=1 to m3 result in a matrix**ac**with m2 rows and m3 columns which shall be called the**employment matrix**.

- In the case of full employment applies

- Let us assume that there is a needs-based national economy in which every company can indicate the volume of employment it needs in each profession in order to provide its production. In order to supply this needs-based producing economy with needs-based and profession-specific work performance a needs-based performance structure can be specified by a
**vector db0 of a needs-based performance structure**which components**db0**indicate for each profession j3 which profession-specific performance according to the profession's requirement must be provided._{j3}

Now another vector shall be introduced as a test criterion for the needs-conformity of the employment structure.

For the sake of simplicity in case of possible economic growth it is assumed that the proportions of the need for profession-specific performances will not change at short notice. This means that every multiple of this vector also results in a needs-based performance structure.

Thus all primary parameters for describing an employment structure have been introduced.

**Secondary parameters:**

All secondary parameters can be derived from the primary ones. They serve to make it easier to evaluate the later results of an optimization.

For each profession j3 the **normalized productivity pn _{j3}** is defined by

This value can be interpreted as the mean productivity of all workers, including those who do not work in this profession. These values would also result in the mean productivity of all workers employed in the profession if the number of workers is very large and the distribution of workers among the respective professions were completely left to chance and their respective skills are not taken into account at all. Thus

The m3 parameters pn_{j3} are combined to a vector **pn**, which represents the **normalized, profession-independent productivity profile** of society.

Using the employment matrix ac and the productivity matrix p, for each profession can be calculated j3 which **work performance d _{j3}** is currently achieved in this profession.

The vector **d**, consisting of the m3 components **d _{j3}**, results in the

The current numbers **ae _{j3}** of workers who are employed in the respective profession j3 are calculated from the employment matrix ac by

They are combined in the vector **ae**

For each profession the **mean productivity pm _{j3}** is defined j3 by

Thus pm_{j3} is the mean productivity of the workers actually employed in this profession, which means it is profession-dependent. (Compare indicative value pm with indicative value pn)

The m3 parameters pm_{j3} are combined to form a vector **pm** which represents the **mean profession-dependent productivity profile** of society. With this vector the "societally necessary working time" for a product is given in the Marxian sense.

The ratio between the mean profession-dependent productivity pm_{j3} and the normalized (mean profession-__in__dependent) productivity pn_{j3} results in the factor fp_{j3} for the increase of productivity in the respective profession j3.

(8) fp_{j3} _{def}= pm_{j3 }/pn_{j3}

The vector db0 is initially given arbitrarily with regard to the level of its components, since at the moment only the relationships of the components to one another are important for the description of a needs-based performance structure. This vector is normalized by introducing a vector **dbn** for a **normalized, needs-based performance structure**.

Since the value of the denominator in this formula is constant in relation to the index j3, it is obvious that the vector dbn represents a multiple (or a fraction) of the vector db0 and is therefore also a needs-based performance structure. In addition, the vector dbn corresponds to the performance structure of a needs-based producing society that consists only of one "normal" worker, whose individual productivity profile is pn.

The vector d of the current societal performance structure due to the employment matrix ac does not have to be needs-based. **The largest vector it contains which represents a needs-based performance structure** is referred to as **db**. db is then the largest vector of a needs-based performance-oriented structure that is contained in vector d, if for all **db _{j3}** with j3= 1 to m3 the following applies:

(10) db_{j3} = fg ×
a ×dbn_{j3}

where for all applies db_{j3 } £
d_{j3} and for at least one j3 applies db_{j3} = d_{j3} .

Hereby the factor **fg** was introduced as well. It represents the **factor of increased overall societal performance** due to the employment matrix ac. It is the most important parameter for evaluating the quality of the employment matrix. Therefore it is used as an objective function of an optimization calculation later on.

The factors **fb _{j3}** for evaluating the needs-conformity of the work performances can be defined by

(11) fb_{j3} _{def}= db_{j3} / d_{j3}

Accordingly, for each profession they indicate how large the share of needs-based work is in relation to the total work performed in the respective profession. The aim for all fb_{j3} is to have the value one, because then work is carried out completely needs-based ably.

From these factors the more complex factor **fbg** can be specified. It provides a quick overview of the needs-conformity of overall societal work performance.

If fbg=1, then in total work is done according to needs. The factor indicates what proportion of workers is involved in the overall societal needs-conform work performance structure db within the possibly non-needs-conform work performance structure d.

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