Progressive bracketed taxation

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In QuartaPiStan the currency is the silver nose-handkerchief (FzA). Linus Quadrangulus earns $150$ FzA per year, Pina Triangula earns $200$. The progressive tax rates are bracketed:

Income bracket (FzA)	Rate
$0-30$	$0\%$
$30-80$	$10\%$
$80-130$	$20\%$
$130-180$	$30\%$
over $180$	$40\%$

• (a) Find how much tax Linus pays and his net salary; do the same for Pina.
• (b) Change the $20\%$ rate so that Linus pays exactly $17\%$ in taxes.

(In the original test the last table row reads “over 240 -> 40%”: that is a typo; the top bracket starts at 180.)

Solution

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

(a) Linus ($150$ FzA):

• $0-30$: $0$
• $30-80$ (50 FzA): $10\% \to 5$
• $80-130$ (50 FzA): $20\% \to 10$
• $130-150$ (20 FzA): $30\% \to 6$

Tax $=5+10+6=21$ FzA; net $=150-21=129$ FzA; rate $=\dfrac{21}{150}=14\%$.

Pina ($200$ FzA): add to the first contributions ($5+10$, then $130-180$, 50 FzA at $30\%\to 15$) the slice $180-200$ (20 FzA) at $40\%\to 8$: $\text{tax}=5+10+15+8=38\ \text{FzA};\quad \text{net}=200-38=162\ \text{FzA};\quad \frac{38}{200}=19\%.$

(b) Let $a$ be the new rate of the $80-130$ bracket. For Linus that portion is $50$ FzA, while $5$ (from $10\%$) and $6$ (from $30\%$) stay fixed: $\text{tax}=5+50a+6=11+50a.$ Imposing $\text{tax}=17\%\cdot 150=25.5$: $11+50a=25.5\ \Rightarrow\ a=\frac{14.5}{50}=0.29=29\%.$

$\boxed{\text{Linus: }21\,/\,129;\quad \text{Pina: }38\,/\,162;\quad a=29\%}$