User:Est. 2021/sandbox/formula

Data

• $\operatorname {arsinh} =\sinh ^{-1}$
• $S=3068$

Legend

• $R$ = Gross personal income
• $N0$ = $Z_{R}$ = Tax rate limit (%)
• $N1$ = Income tax threshold + Tax deductions (eg. 1€/h = 24€/d = c. 730€/m = c. 8760€/y)
• $N2$ = Social dividend (eg. 0,50€/h = 12€/d = c. 365€/m = c. 4380€/y)
• $B$ = Taxable income = if $R>N1$, then $B=R-N1$, otherwise $0$
• $T$ = Unlimited income tax = if $B>0$, then $T=B*X_{B}=R*X_{R}$, otherwise $0$
• $TZ$ = Limited income tax = if $B>0$, then $TZ=B*XZ_{B}=R*XZ_{R}$, otherwise $0$
• $Y$ = Net personal income = ${\begin{cases}R-T,&{\text{if unlimited}}\\R-TZ,&{\text{if limited}}\end{cases}}$
• $P$ = Spending power = $Y+N2$

Notes

• $N1_{A},N2_{A},R_{A},B_{A},Y_{A},P_{A}$ = Annual... (default)
• $N1_{M},N2_{M},R_{M},B_{M},Y_{M},P_{M}$ = Monthly...

Formula

• $X_{B}=1-{\frac {\operatorname {arsinh} \left({\sqrt {{\frac {1}{4}}+{\sqrt {1+\left({\frac {B}{S}}\right)^{2}}}-1}}-{\frac {1}{2}}\right)}{{\sqrt {{\frac {1}{4}}+{\sqrt {1+\left({\frac {B}{S}}\right)^{2}}}-1}}-{\frac {1}{2}}}}$
• $X_{R}=X_{B}*{\frac {B}{R}}$

Limited Tax — Method 1

• $XZ_{R}=X_{R}*Z_{R}=XZ_{B}*{\frac {B}{R}}=TZ:R$
• $XZ_{B}=X_{B}*Z_{B}=XZ_{R}:{\frac {B}{R}}=TZ:B$
• $Z_{B}={\frac {XZ_{B}}{X_{B}}}=Z_{R}$

Limited Tax — Method 2

• $XZ_{R}=$ if $X_{R}<Z_{R}$, then $X_{R}$, otherwise $Z_{R}$
• $XZ_{B}=XZ_{R}:{\frac {B}{R}}$
• $Z_{B}={\frac {XZ_{B}}{X_{B}}}\geq {Z_{R}}$

Notes to $Z_{B}$ — Method 2

• if $X_{R}<Z_{R}$, then $Z_{B}=100\%$
• if $Z_{R}=100\%$, then $Z_{B}=100\%=Z_{R}$, otherwise $Z_{B}>Z_{R}$

Footnotes