Progressive taxation versus flat tax

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In the Viking enclave of Sjugmärsstørd the currency is Odin’s axe (AdO). Nobles pay a flat tax of $20\%$; peasants pay a progressive bracketed tax: $0\%$ from $0$ to $3$ AdO; $10\%$ from $3$ to $10$; $25\%$ from $10$ to $23$; $35\%$ from $23$ to $35$; $55\%$ above $35$. Alfarin is a noble with a gross salary of $60$ AdO; Eyvald is a peasant with gross income of $30$ AdO per year.

• (a) find the tax paid by Eyvald and his net monthly income (over $12$ months);
• (b) decide whether Eyvald’s average tax rate is higher or lower than Alfarin’s flat rate.

Solution

(a) Progressive taxation applies each rate only to the portion of income falling in the corresponding bracket. With income $30$ AdO: $T=\underbrace{0\%\cdot 3}_{0}+\underbrace{10\%\cdot(10-3)}_{0.7}+\underbrace{25\%\cdot(23-10)}_{3.25}+\underbrace{35\%\cdot(30-23)}_{2.45}=6.40\ \text{AdO}.$ Net yearly $=30-6.40=23.60$ AdO; monthly $=\dfrac{23.60}{12}\approx1.97$ AdO.

(b) Eyvald’s average rate is $\dfrac{6.40}{30}\approx21.3\%$, higher than Alfarin’s flat $20\%$. $\boxed{\ T_{\text{Eyvald}}=6.40\ \text{AdO};\quad \text{net monthly}\approx1.97\ \text{AdO};\quad \bar\tau\approx21.3\%>20\%\ }$