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\title{SEE: Detail of Wealth Distribution calculation}
\author{Willy H. Gerber}

\begin{document}

\title{Detail of Wealth Distribution calculation}

People number

\begin{equation}
N = \int_{\varepsilon_{min}}^{\varepsilon_{max}} \frac{1}{e^{-\beta\varepsilon + \alpha} - 1} \frac{d\varepsilon}{\varepsilon_{max} - \varepsilon_{min}}
\end{equation}

Integration formula

\begin{equation}
I = \int \frac{dx}{p + q e^{ax}} = \frac{x}{p} - \frac{1}{ap} \ln(p + q e^{ax})
\end{equation}

Replacement
\begin{equation}
p = -1
\end{equation}
\begin{equation}
q = e^{\alpha}
\end{equation}
\begin{equation}
a = -\beta
\end{equation}
\begin{equation}
x = \varepsilon
\end{equation}

\begin{equation}
I = \int \frac{d\varepsilon}{-1 + e^{\alpha} e^{-\beta\varepsilon}} = \frac{\varepsilon}{-1} - \frac{1}{\beta} \ln(-1 + e^{\alpha} e^{-\beta\varepsilon})
\end{equation}

\begin{equation}
I = \int_{\varepsilon_{min}}^{\varepsilon_{max}} \frac{d\varepsilon}{-1 + e^{\alpha} e^{-\beta\varepsilon}} = -\varepsilon_{max} - \frac{1}{\beta} \ln(e^{\alpha} e^{-\beta\varepsilon_{max}} - 1) + \varepsilon_{min} + \frac{1}{\beta} \ln(e^{\alpha} e^{-\beta\varepsilon_{min}} - 1)
\end{equation}

\begin{equation}
I = -(\varepsilon_{max}-\varepsilon_{min}) - \frac{1}{\beta} \ln(e^{\alpha} e^{-\beta\varepsilon_{max}} - 1) + \frac{1}{\beta} \ln(e^{\alpha} e^{-\beta\varepsilon_{min}} - 1)
\end{equation}

\begin{equation}
I = - (\varepsilon_{max}-\varepsilon_{min}) - \frac{1}{\beta} \ln\frac{e^{\alpha}e^{-\beta\varepsilon_{max}} - 1}{e^{\alpha} e^{-\beta\varepsilon_{min}} - 1}
\end{equation}

\begin{equation}
N = \frac{I}{\varepsilon_{max}-\varepsilon_{min}}
\end{equation}

\begin{equation}
N = -1 - \frac{1}{\beta(\varepsilon_{max}-\varepsilon_{min})} \ln\frac{e^{\alpha}e^{-\beta\varepsilon_{max}} - 1}{e^{\alpha}e^{-\beta\varepsilon_{min}} - 1}
\end{equation}

\begin{equation}
-(N + 1)\beta(\varepsilon_{max}-\varepsilon_{min}) = \ln\frac{e^{\alpha}e^{-\beta\varepsilon_{max}} - 1}{e^{\alpha}e^{-\beta\varepsilon_{min}} - 1}
\end{equation}

\begin{equation}
e^{-(N + 1)\beta(\varepsilon_{max}-\varepsilon_{min})} = \frac{e^{\alpha}e^{-\beta\varepsilon_{max}} - 1}{e^{\alpha}e^{-\beta\varepsilon_{min}} - 1}
\end{equation}

\begin{equation}
e^{-(N + 1)\beta(\varepsilon_{max}-\varepsilon_{min}) - \beta\varepsilon_{min}}e^{\alpha} - e^{-(N + 1)\beta(\varepsilon_{max}-\varepsilon_{min})} = e^{\alpha}e^{-\beta\varepsilon_{max}} - 1
\end{equation}

\begin{equation}
1 - e^{-(N + 1)\beta(\varepsilon_{max}-\varepsilon_{min})} = e^{\alpha}e^{-\beta\varepsilon_{max}} - e^{-(N + 1)\beta(\varepsilon_{max}-\varepsilon_{min}) - \beta\varepsilon_{min}}e^{\alpha}
\end{equation}

\begin{equation}
e^{\alpha} = \dfrac{1 - e^{-(N + 1)\beta(\varepsilon_{max}-\varepsilon_{min})}}{e^{-\beta\varepsilon_{max}} - e^{-(N + 1)\beta(\varepsilon_{max}-\varepsilon_{min}) - \beta\varepsilon_{min}}}
\end{equation}

\begin{equation}
e^{\alpha} = e^{\beta\varepsilon_{max}} \dfrac{1 - e^{-(N + 1)\beta(\varepsilon_{max}-\varepsilon_{min})}}{1 - e^{-N\beta(\varepsilon_{max}-\varepsilon_{min})}}
\end{equation}

\begin{equation}
e^{\alpha} = e^{\beta\varepsilon_{max}} \gamma
\end{equation}

\begin{equation}
\gamma = \dfrac{1 - e^{-(N + 1)\beta(\varepsilon_{max}-\varepsilon_{min})}}{1 - e^{-N\beta(\varepsilon_{max}-\varepsilon_{min})}}
\end{equation}


Case $N \gg 1$ then

\begin{equation}
e^{\alpha} = \dfrac{1}{e^{-\beta\varepsilon_{max}}}
\end{equation}

\begin{equation}
\alpha = \beta\varepsilon_{max}
\end{equation}

Distribution

\begin{equation}
I(\varepsilon) = \int_{\varepsilon_{min}}^{\varepsilon} \frac{d\varepsilon'}{-1 + e^{\beta(\varepsilon_{max}-\varepsilon')}} = -\varepsilon - \frac{1}{\beta} \ln(\gamma e^{\beta(\varepsilon_{max}-\varepsilon)} - 1) + \varepsilon_{min} + \frac{1}{\beta} \ln(\gamma e^{\beta(\varepsilon_{max}-\varepsilon_{min})} - 1)
\end{equation}

\begin{equation}
P(\varepsilon) = \frac{1}{\beta(\varepsilon_{max}-\varepsilon_{min})} \ln \dfrac{\gamma e^{\beta(\varepsilon_{max}-\varepsilon_{min})} - 1}{\gamma e^{\beta(\varepsilon_{max}-\varepsilon)} - 1} - \dfrac{\varepsilon-\varepsilon_{min}}{\varepsilon_{max}-\varepsilon_{min}}
\end{equation}

\begin{equation}
\Delta = \dfrac{\gamma e^{\beta(\varepsilon_{max}-\varepsilon_{min})} - 1}{\gamma e^{\beta(\varepsilon_{max}-\varepsilon)} - 1} = \dfrac{(1 - e^{-(N + 1)\beta(\varepsilon_{max}-\varepsilon_{min})}) e^{\beta(\varepsilon_{max}-\varepsilon_{min})} - 1 + e^{-N\beta(\varepsilon_{max}-\varepsilon_{min})}}{(1 - e^{-(N + 1)\beta(\varepsilon_{max}-\varepsilon_{min})})e^{\beta(\varepsilon_{max}-\varepsilon)} - 1 + e^{-N\beta(\varepsilon_{max}-\varepsilon_{min})}}
\end{equation}

\begin{equation}
\Delta = \dfrac{e^{\beta(\varepsilon_{max}-\varepsilon_{min})} - e^{-N\beta(\varepsilon_{max}-\varepsilon_{min})} - 1 + e^{-N\beta(\varepsilon_{max}-\varepsilon_{min})}}{(1 - e^{-(N + 1)\beta(\varepsilon_{max}-\varepsilon_{min})})e^{\beta(\varepsilon_{max}-\varepsilon)} - 1 + e^{-N\beta(\varepsilon_{max}-\varepsilon_{min})}}
\end{equation}

\begin{equation}
\Delta = \dfrac{e^{\beta(\varepsilon_{max}-\varepsilon_{min})} - 1}{(1 - e^{-(N + 1)\beta(\varepsilon_{max}-\varepsilon_{min})})e^{\beta(\varepsilon_{max}-\varepsilon)} - 1 + e^{-N\beta(\varepsilon_{max}-\varepsilon_{min})}}
\end{equation}

Limes $\varepsilon \rightarrow \varepsilon_{max}$

\begin{equation}
D = \dfrac{e^{\beta(\varepsilon_{max}-\varepsilon_{min})} - 1}{- e^{-(N + 1)\beta(\varepsilon_{max}-\varepsilon_{min})} + e^{-N\beta(\varepsilon_{max}-\varepsilon_{min})}}
\end{equation}

\begin{equation}
D = \dfrac{e^{\beta(\varepsilon_{max}-\varepsilon_{min})} - 1}{e^{-(N + 1)\beta(\varepsilon_{max}-\varepsilon_{min})}(- 1 + e^{\beta(\varepsilon_{max}-\varepsilon_{min})})}
\end{equation}

\begin{equation}
D = e^{(N + 1)\beta(\varepsilon_{max}-\varepsilon_{min})}
\end{equation}

\begin{equation}
P(\varepsilon_{max}) = \frac{1}{\beta(\varepsilon_{max}-\varepsilon_{min})} \ln D - 1
\end{equation}

\begin{equation}
P(\varepsilon_{max}) = N
\end{equation}

Aproximation

\begin{equation}
\Delta = \varepsilon - \varepsilon_{min}
\end{equation}

\begin{equation}
P(\Delta) = \frac{1}{\beta(\varepsilon_{max}-\varepsilon_{min})} \ln \dfrac{\gamma e^{\beta(\varepsilon_{max}-\varepsilon_{min})} - 1}{\gamma e^{\beta(\varepsilon_{max}-\varepsilon_{min} - \Delta)} - 1} - \dfrac{\Delta}{\varepsilon_{max}-\varepsilon_{min}}
\end{equation}

\begin{equation}
P(\Delta) = \frac{1}{\beta(\varepsilon_{max}-\varepsilon_{min})} \left( \ln(\gamma e^{\beta(\varepsilon_{max}-\varepsilon_{min})} - 1) - \ln(\gamma e^{\beta(\varepsilon_{max}-\varepsilon_{min} - \Delta)} - 1)\right) - \dfrac{\Delta}{\varepsilon_{max}-\varepsilon_{min}}
\end{equation}

\begin{equation}
\ln{(\gamma e^{\beta(\varepsilon_{max}-\varepsilon_{min} - \Delta)} - 1)} \sim \ln{(\gamma e^{\beta(\varepsilon_{max}-\varepsilon_{min})} - 1)} - \dfrac{\beta \gamma e^{\beta(\varepsilon_{max}-\varepsilon_{min})}}{\gamma e^{\beta(\varepsilon_{max}-\varepsilon_{min})} - 1} \Delta
\end{equation}

\begin{equation}
P(\Delta) \sim \frac{1}{(\varepsilon_{max}-\varepsilon_{min})} \left(\dfrac{\gamma e^{\beta(\varepsilon_{max}-\varepsilon_{min})}}{\gamma e^{\beta(\varepsilon_{max}-\varepsilon_{min})} - 1} \Delta\right) - \dfrac{\Delta}{\varepsilon_{max}-\varepsilon_{min}}
\end{equation}

\begin{equation}
P(\Delta) \sim \left(\dfrac{\gamma e^{\beta(\varepsilon_{max}-\varepsilon_{min})}}{\gamma e^{\beta(\varepsilon_{max}-\varepsilon_{min})} - 1} - 1\right) \dfrac{\Delta}{\varepsilon_{max}-\varepsilon_{min}}
\end{equation}

\begin{equation}
P(\Delta) \sim \left(\dfrac{1}{\gamma e^{\beta(\varepsilon_{max}-\varepsilon_{min})} - 1}\right) \dfrac{\Delta}{\varepsilon_{max}-\varepsilon_{min}}
\end{equation}

\begin{equation}
\ln(a e^{-\Delta} - 1) = \ln(a - 1) - \dfrac{a e^{-\Delta}}{a e^{-\Delta} - 1}\mid_{\Delta=0} \Delta + \dfrac{ - a e^{-\Delta} (a e^{-\Delta} - 1) + a e^{-\Delta} a e^{-\Delta}}{2(a e^{-\Delta} - 1)^2}\mid_{\Delta=0} \Delta^2
\end{equation}

\begin{equation}
\ln(a e^{-\Delta} - 1) = \ln(a - 1) - \dfrac{a}{a - 1} \Delta + \dfrac{a}{2(a - 1)^2} \Delta^2
\end{equation}


\end{document}