Science:Math Exam Resources/Courses/MATH100 C/December 2024/Question 12 (d)/Solution 1

From part (a), we know that at $x=1$ (i.e., $10{,}000), the marginal tax rate is $$T^{\prime }(1)=\frac{5}{9}\approx 0.556.$$

Thus, a middle-income earner pays about 55.6 cents in tax for each additional dollar earned.

From part (c), for very large incomes, $$\underset{x\to \mathrm{\infty }}{\lim }\frac{T(x)}{x}=\frac{1}{2}.$$

So high-income earners pay about 50 cents in tax per dollar of income on average.

Therefore, a middle-income earner pays a higher tax rate on each additional dollar than a high-income earner pays on average, which provides evidence that the system is less favorable to middle-income earners.

(Note: Although $\frac{T(x)}{x}$ increases with $x$, meaning average tax rises with income, the marginal tax rate at middle incomes is higher than the average rate for high incomes.)