Statistical Sociology
by Dr. Willy H. Gerber



Abstract

Based on the mathematical methods used in the description of multiple particle system in Physics, I develop a model that descript how a society will distribute resources (Lorenz Curve, Gini factor), create unemployment and low level of resources can trigger and sustained mass movements.

Because that particular method is called in Physics "Statistical Mechanics" I name this approach "Statistical Sociology".



1. Working hypothesis

For building the model, I state two basic hypothesis on which the model structure is based.

The first one has to do with the parameters we need to describe the state of one individual in particular and society as a whole.

Hypothesis 1: The state of a person can be describe by a scalar measurable quantity "ω" we will call "well-being".

Each person in our society has a particular level of well-being. To this quantity contributes not only material things like the earning and savings of the person, also the social acceptance and other factors necessary to succeed in society.

One of the key results of this model is to estimate how this individual "well-being" is distribute among the different people in this society. The quantity could be concentrated only in few or equally around the system. Each situation is a possible state of our society. The second hypothesis will state that we are not defining a "a priory" distribution:

Hypothesis 2: Each possible state of the society is equally alike.

Starting with this basic hypothesis and few simple assumptions we hope to describe the behaviour of a society.


2. Counting States

Suppose that the society has a limited amount of well-being W to be distribute among there members. In case of the material needs, it could be in a first approximation the Gross Domestic Product (GDP). The condition of a fix maximal Well-being for the society implies that the sum of all individual well-being of each person i must be:

(2.1)

The question now is how probable it is, that a single person has a well-being level between ω and ω+dω. By definition of probability, this will be the fraction of cases where the person has the level ω to the total number of situations we can find our society. If Ω( ω) is the number of states of the society where the person has the well-being level ω and Ω is the total number of states the society can be in, the probability will read:

(2.2)

To calculate the number of states Ω, I assume that the total amount of well-being W can be split in

(2.3)

little segments of length Δ. I have now to distribute the N persons in this different cells assuring that at all time the condition (2.1) is fulfilled. I start with the simple case of only one persons. In this case there are only 1 alternative because the person has a well-being level equal to the total amount of well-being in the system.

(2.4)

In case of two persons one can be in any of the M cells. The second can only be in the cell with the level of well-being that satisfies the condition (2.2) and there fore the number of states is

(2.5)

I consider now the case with three persons. If I suppose that the third person has a well-being of ω, the other two persons can be in

(2.6)

possible states with a total well-being of the remaining W-ω. Since the third person can occupy any of the M cells, the total number of states will be

(2.7)

In a similar form I can calculate the case with four persons getting the number of states:

(2.8)

If I repeat this exercise N times I will get the number of possible states for N persons to be

(2.9)

In this case is M >> N and also N >> 1, so that I can rewrite the coefficient

(2.10)

and get in a very good approximation for the number of states of N person with a total well-being of W is

(2.11)

With this distribution I can now analyse how the well-being is distributed N personas in a society.


3. The Lorenz curve and Gini factor for the idealized case

The number of states for the case one person has a well-being level of ω is equal to the number of states for only N-1 persons with a total well-being of W-ω:

(3.1)

Inserting (2.11) and (3.1) into (2.2) I get for the probability:

(3.2)

If I introduce the average well-being as the total well-being divide by the number of people in the society:

(3.3)

Using the relation

(3.4)

I get in case of a large number of persons (N >>1) the expression:

(3.5)

This is the idealized case where I have assumed that each of the N person has free access to well-being. What are the implications of (3.4) for a Society?.

With (3.5) I can sketch a Lorenz curve of the cumulative distribution function of well-being in function of associated cumulate population. I can define a cumulated population function integrating (3.4) over well-being up to a fraction µ of the total amount W:

(3.6)

In a similar form I introduce a cumulated well-being integrating population density (3.5) multiplying by the level of well-being:

(3.7)

With a partial integration the express of cumulated well-being can be writen in terms of the cumulated population getting an expression for the Lorenz curve:

(3.8)

The graphic representation of this equation shows the classical curve:


The Gini factor can be easily calculate sustracting from the surface under the curve γ=µ the integral of (3.2):

(3.9)

This will be the Gini factor for a society in wich each member of the society has an equal opportunity to access well-being. In section 5. I will descusse the case where not everybody has access to well-being.


4. The Partition Function

The principal result of section 3 is that the probability to find a person r with a well-being level ωr between ω and ω + dω is described by :

(4.1)

where β must be chosen in a way that the total well-being will be equal W

(4.2)

This means that I have

(4.3)

persons with a well-being level of ωr and the average well-being will be given by the expression

(4.4)

I introduce the factor β because this expression can be rewritten in the following way:

(4.5)

introducing a function Z defined by

(4.6)

The function Z is called in Physics the "Partition Function" (or in German "Zustandssumme") and will play a key role in the modelling. The Partition function can be use to calculate all moments of the distribution; for example the quadratic mean will read

(4.7)

and the variance of the system

(4.8)

This schema delivers the same result I already develop in section 3; if I suppose that the possible well-being states is any positive value I will have

(4.9)

and I can calculate β for N >> 1

(4.10)

With this β I get the distribution (3.4). If I assume that I have a legal minimum wages of ω0, the Partition Function (4.1) will read

(4.11)

the mean wages be

(4.12)

and the Distribution function is

(4.13)

To check this first result I study the OES statistics for the US market in 2005 [1] considering the procentage of people earning a yearlly wages in ranges 0-10k$, 10-20k$, 20-30k$ etc. (k$ = 1000 USD). From this data we got a mean wages of 37.88 k$ and a first aproximation by (4.10) using a minimum yearlly wages of 13.68 k$:


This shows that the Distribution (4.13) may not be perfect but shows the right tendency in the full wages range. This means tat the haul wages Distribution can be modeled in this general way despide the fact that for the individual position-wages detail relation a modeling of the offer and demnd will be necessary.


5. The Grand Partition Function

In my first model I assume that the number of members where N, despite the fact that some may not have income at all (unemployed). In case the number of members is n < N the number of states is given by

(5.1)

and the probability that a person earns ω in a System where n people of N get well-being is

(5.2)

In (4.1) I simplify the expresion for the probability introducing a factor β beeing in the simple case N/W. In this case the number of persons will vary and can not be included in β. We also have to take into account that we have to determine the expected number of persons getting well-being. For both reasons and following the steps to set (4.1) y will generalize these assumption writing the Ansatz:

(5.3)

with a person number of

(5.4)

and the conditions:

(5.5)

and:

(5.6)

In this case the expectet well-being will be

(5.7)

and the mean number of pesons participating in the system

(5.8)

In analogy to (4.6) I will introduce the more general Partition Function

(5.9)

and call it the "Grand Partition Function". Similar to (4.5) the expected well-being and the number of persons can be calculated from the Grand Partition Function:

(5.10)

(5.11)

Because the number n runs from 0 to a very large number N, I can add the Grand Partition Function in n using the geometric series to get

(5.12)

Using (5.10) I now can estimate the average number of persons with a well-being level of ω

(5.13)

Applying (5.8) to the Grand Partition Function (5.10) I get the modify Well-being distribution:

(5.12)

In a similar way to the first sketch for the ideal case I again show the distribution of well-being (income) for the OES US Wages data from 2005:


Equation (4.9) can be add on the number of persons per state simplifying the Partition function to

(4.10)

or in the logarithmic form

(4.11)

The partition function can be used to calculate the mean value

(4.12)

the quadratic mean

(4.13)

and the variance of the system

(4.14)

Now we can study the more realistic case implied by the partition function (4.10).


5. Well-being distribute in a society

Before we can estimate the Lorenz curve for this case, we must calculate the factor α to establish that the number of persons witch have access to well-being is less-equal the total number of persons N:

(5.1)

If we also assume that the mayor contribution to the well-being level, is due to the wages the persons are receiving, we could introduce a minimal wages ω0. In this case we can calculate the number of persons with access to well-being applying (4.12) to the partition function (4.13) in the continuous approximation (4.6):

(5.2)

After integration we get an equation for the factor α:

(5.3)

References

[1] Download Occupational Employment and Wage Estimates
# National 3-digit NAICS Industry-Specific estimates (1,987 KB)
http://www.bls.gov/oes/oes_dl.htm#2005_m