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Income model
Storyboard 
Based on the concept by Peter Turchin (1), which suggests that society can be divided into two main segments, we can distinguish between the first, which we will call the "population," and the second, known as the "elite." According to this perspective, the elite exerts control over the population and exploits its resources, which can lead to tensions and attempts by the elite to curtail the freedoms of the population.
However, it is also observed that when the number of individuals in the elite group grows disproportionately, competition for available resources ensues. This rivalry can lead to the immobilization of the population, effectively utilized to serve the objectives of the elite. Under these circumstances, there is a risk of destabilization within the social structure, a situation that can only be alleviated by reducing the number of elite members.
The approach presented here builds upon the ideas put forth by Turchin in his work "Historical Dynamics" (Princeton University Press, 2019) and proposes an alternative model based on the analysis of socio-physical interactions.
(1) Turchin, Peter. "Historical Dynamics." Princeton University Press, 2019.
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The concept of the model
Description
Two complementary themes interplay here. On one hand, there's the existing socio-economic structure at a given moment, which we can refer to as the current state of the system. On the other hand, we have the evolution of these factors over time. In Turchin's model, the initial conditions are defined, and Lotka-Volterra type equations are established to elucidate how parameters vary based on their interactions.
In this case, we're shaping a distinct framework. Socio-economic variables are introduced and their linkage with economic and demographic parameters is established, allowing for the estimation of their values for each country. Subsequently, evolutions are considered using both economic and demographic models of the situation, ultimately leading to an estimation of the trajectory of the key variables within the model.
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Model Populations
Equation
The society is modeled as consisting of two groups. The first is the ordinary population, denoted by $n_p$. Added to this is the number of individuals comprising the elite, which we'll refer to as $n_e$. The sum of these two must equal the total number of individuals in the population, denoted as $N$.
 $ n_p + n_e = N $ |
To simplify the modeling process, we will assume that these numbers represent the respective total populations, including the families of those who participate in societal activities, such as working, managing, or investing.
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Income in the model
Equation
Each social group has its income, which corresponds to the portion of the gross domestic product they have access to, regardless of how they acquire it. If the total gross domestic product is $E$, and the average income per person in the population is $\epsilon_p$, while for the elite it's $\epsilon_e$, we must have in total
| $ n_p \epsilon_p + n_e \epsilon_e = E $ |
This is a simplification, as individual incomes vary within both groups. However, they should be understood as average values, and this simplification will actually allow us to estimate levels based on other known economic and demographic data.
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Fraction of the population
Equation
The fraction of the population of each type, with $i=p$ for the ordinary population and $i=e$ for the elite, is defined in terms of the number of each group $n_i$ and the total $N$, using the formula:
| $ p_i = \displaystyle\frac{ n_i }{ N }$ |
This approach will simplify the solution of the equations.
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Fraction of income
Equation
The distribution of income for each segment of the population, with $i=p$ for the ordinary population and $i=e$ for the elite, is defined in terms of the number of individuals in each group $n_i$, the income of each group $\epsilon_i$, and the gross domestic product $E$, through the formula:
| $ q_i = \displaystyle\frac{ n_i \epsilon_i }{ E }$ |
This approach will aid in simplifying the resolution of the equations.
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Normalization of the fraction of the population
Equation
The normalization condition for the population fractions $p_p$ for the ordinary population and $p_e$ for the elite is as follows:
| $ p_p + p_e = 1 $ |
Since the sum of the populations is equal to the total number,
| $ n_p + n_e = N $ |
we obtain by dividing this expression by the total number $N$ and using the fraction definition,
| $ p_i = \displaystyle\frac{ n_i }{ N }$ |
the resulting expression is
| $ p_p + p_e = 1 $ |
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Normalization of the fraction of income
Equation
The normalization condition for the income fractions $q_p$ for the common population and $q_e$ for the elite is as follows:
| $ q_p + q_e = 1 $ |
Since the sum of the populations is equal to the total number,
| $ n_p + n_e = N $ |
we obtain by dividing this expression by the total number $N$ and using the fraction definition,
| $ p_i = \displaystyle\frac{ n_i }{ N }$ |
the resulting expression is
| $ q_p + q_e = 1 $ |
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Gini coefficient
Description
The Gini coefficient is a measure that evaluates economic equity within a society by providing information about the income distribution among its population. To calculate it, the Lorenz curve is first plotted, illustrating how incomes accumulate relative to the population ordered from the lowest to the highest income levels.
In the ideal case of perfect equality, where everyone has the same income, the Lorenz curve would be a straight diagonal line on the graph representing the fraction of the population versus the fraction of accumulated income. However, in societies with high levels of inequality, the Lorenz curve deviates from the diagonal and curves towards the axis of accumulated income fraction, ascending more steeply towards the end.
The Gini coefficient is calculated as the ratio of the area between the diagonal and the Lorenz curve (represented as A) to the area under the line representing perfect income equality (represented as A+B). Mathematically, it is expressed as:
$Gini = \displaystyle\frac{A}{A + B}$
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The Gini of the model
Equation
If we consider a society with only two income levels, the Gini diagram simplifies to the red and green lines depicted in the following illustration:
In this diagram, the proportions of the population are represented by $p_p$ and $p_e$, while the proportions of income are represented by $q_p$ and $q_e$. This setup enables the calculation of areas to estimate the Gini coefficient, which can be derived as follows:
| $ Gini = q_e - p_e $ |
The area B, as indicated in the graph, is equal to the sum of the triangles with the colored hypotenuses plus the rectangle in the bottom right corner:
$B = \displaystyle\frac{1}{2}p_nq_n + p_eq_n + \displaystyle\frac{1}{2}p_eq_e$
The area A and B are equal to the triangle below the line of equal society, which is equal to 1/2:
$A + B = \displaystyle\frac{1}{2}$
therefore, the area of A is
$A = A + B - B = \displaystyle\frac{1}{2} - \displaystyle\frac{1}{2}p_nq_n - p_eq_n - \displaystyle\frac{1}{2}p_eq_e$
Therefore, the Gini coefficient is
$Gini = \displaystyle\frac{A}{A+B}=1-p_nq_n-2p_eq_n-p_eq_e$
which, with
| $ p_p + p_e = 1 $ |
and
| $ q_p + q_e = 1 $ |
reduces to
| $ Gini = q_e - p_e $ |
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Another Gini ratio of the model
Equation
Another relationship that can be derived for the two-level income model is obtained from
| $ Gini = q_e - p_e $ |
which, through normalization equations, can be expressed using the respective fractions of the population $p_p$ and their incomes $q_p$:
| $ Gini = p_p - q_p $ |
If we substitute the fractions of the elite $p_e$ and their incomes $q_e$ in
| $ Gini = q_e - p_e $ |
with those of the population obtained from the normalization relationships
| $ p_p + p_e = 1 $ |
and
| $ q_p + q_e = 1 $ |
we obtain
| $ Gini = p_p - q_p $ |
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Canonical distribution for income
Equation
In statistical mechanics, the canonical distribution is introduced to estimate the number of particles $n(\epsilon)$ occupying an energy state $\epsilon$ using the equation:
$n(\epsilon) d\epsilon = C e^{-\beta\epsilon} d\epsilon$
Here, the factor $\beta$ is related to the temperature $T$ as follows:
$\beta = \displaystyle\frac{1}{k_B T}$
where $k_B$ is the Boltzmann constant.
In econophysics, income is often associated with energy, allowing us to use the same distribution by reinterpreting $\epsilon$ as income. The normalization constant $C$ can be determined so that, given that the total income is $E$ and the total number of individuals is $N$, the distribution of people based on income $\epsilon$ should be:
| $ n(\epsilon) d\epsilon =\displaystyle\frac{ N \beta e^{- \beta \epsilon } d\epsilon }{1- e^{- \beta E }}$ |
The significance of $\beta$ or, alternatively, temperature $T$, will become clearer in the subsequent discussion.
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Lorenz curve population factor
Equation
By examining the population distribution based on their incomes, as represented by the equation
| $ n(\epsilon) d\epsilon =\displaystyle\frac{ N \beta e^{- \beta \epsilon } d\epsilon }{1- e^{- \beta E }}$ |
,
it is possible to determine the number of individuals with incomes below a given threshold, denoted as $\epsilon$.
The function $p(u)$ reflects the proportion of people with incomes below $\epsilon$, and this proportion is obtained by dividing the distribution by the total population, $N$:
The function $p(\epsilon)$ indicates the proportion of people with income less than $\epsilon$, which is obtained by dividing the distribution by the total population $N$:
$p(\epsilon) = \displaystyle\frac{\beta}{1- e^{-\beta E}}\displaystyle\int_0^{\epsilon} du e^{-\beta u}$
Upon solving this integral, the equation simplifies to:
| $ p(\epsilon) = \displaystyle\frac{1 - e^{- \beta \epsilon }}{1 - e^{ -\beta E }}$ |
This parameter is crucial in the Lorenz curve of the model and describes the composition of the population in relation to the income distribution.
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Population Factor Equation
Equation
If one evaluates the proportion of the population with incomes equal to or below $\epsilon_p$, it can be estimated using
| $ p(\epsilon) = \displaystyle\frac{1 - e^{- \beta \epsilon }}{1 - e^{ -\beta E }}$ |
As the proportion equals $p_p$, it follows that in the limit of $\beta E\gg 0$, we have
| $ p_p = 1 - e^{- \beta \epsilon_p }$ |
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Lorenz curve income factor
Equation
According to the model, the cumulative income of individuals with incomes below $\epsilon$ can be calculated by integrating the number of individuals:
| $ n(\epsilon) d\epsilon =\displaystyle\frac{ N \beta e^{- \beta \epsilon } d\epsilon }{1- e^{- \beta E }}$ |
weighted by their respective incomes, which is:
$\displaystyle\frac{N\beta}{1-e^{-\beta E}}\displaystyle\int_0^{\epsilon} u e^{-\beta u} du$
By integrating this expression and dividing it by the total income, we obtain the fraction of income for individuals who have an income less than $\epsilon$:
| $ q(\epsilon) = \displaystyle\frac{ N }{ \beta E }\displaystyle\frac{(1-(1+ \beta \epsilon )e^{- \beta \epsilon })}{(1-e^{- \beta E })}$ |
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Calculation of the factor $\beta$
Equation
The distribution given by the equation
| $ q(\epsilon) = \displaystyle\frac{ N }{ \beta E }\displaystyle\frac{(1-(1+ \beta \epsilon )e^{- \beta \epsilon })}{(1-e^{- \beta E })}$ |
when evaluating the in $E$ it must necessarily be equal to the unit. In the limit of a large value of $\beta E$ this implies that:
| $ \beta = \displaystyle\frac{ N }{ E } $ |
This, in turn, signifies that the $\beta$ factor is equal to the inverse of the GDP per capita.
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Income Factor Equation
Equation
If we evaluate the proportion of income corresponding to individuals with incomes equal to or below $\epsilon_p$, we can estimate it using
| $ q(\epsilon) = \displaystyle\frac{ N }{ \beta E }\displaystyle\frac{(1-(1+ \beta \epsilon )e^{- \beta \epsilon })}{(1-e^{- \beta E })}$ |
Since the proportion is equal to $q_p$, it follows that in the limit of $\beta E\gg 0$, we obtain
| $ q_p = 1 - (1+ \beta \epsilon_p )e^{- \beta \epsilon_p }$ |
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Model's Gini
Equation
By utilizing both equations for the factors of individuals and incomes, along with the Gini coefficient for a two-tier society, we arrive at:
| $ Gini = \beta \epsilon_p e^{ \beta \epsilon_p }$ |
By utilizing the equations for the factors
| $ p_p = 1 - e^{- \beta \epsilon_p }$ |
and
| $ q_p = 1 - (1+ \beta \epsilon_p )e^{- \beta \epsilon_p }$ |
we arrive at
| $ Gini = p_p - q_p $ |
which implies that
| $ Gini = \beta \epsilon_p e^{ \beta \epsilon_p }$ |
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