Progressive taxation and calibrating a rate

Text

In the state of Carpistan taxes are paid with these progressive rates: up to $30$ golden coffees $0\%$; from $30$ to $80$ coffees $15\%$; from $80$ to $140$ coffees $25\%$; from $140$ to $200$ coffees $35\%$; over $200$ coffees $45\%$.

• (a) Giovanni Tordi earns $160$ coffees per year. Find the percentage of tax he pays on his gross salary.
• (b) How can you change the $25\%$ rate so that Giovanni pays $50$ coffees of tax per year?

Solution

Progressive taxation: each bracket taxes only the portion of income falling into it.

(a) Giovanni ($160$ coffees):

• $0-30$: $0$
• $30-80$ (50): $15\% \to 7.5$
• $80-140$ (60): $25\% \to 15$
• $140-160$ (20): $35\% \to 7$

Tax $=7.5+15+7=29.5$ coffees; effective rate $=\dfrac{29.5}{160}=0.184375\approx 18.44\%$.

(b) Let $a$ be the new rate of the $80-140$ bracket. For Giovanni that portion is $60$ coffees, while $7.5$ (from $15\%$) and $7$ (from $35\%$) stay fixed: $\text{tax}=7.5+60a+7=14.5+60a.$ Imposing $\text{tax}=50$: $14.5+60a=50\ \Rightarrow\ a=\frac{35.5}{60}=\frac{71}{120}\approx 0.5917=59.17\%.$

$\boxed{\text{tax}=29.5\ (18.44\%);\quad a=\tfrac{71}{120}\approx 59.17\%}$