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

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MATH100 C December 2024

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Question 12 (d)

Consider an income tax system in which the tax $T$ paid on an income $x$ is given by the formula $$T(x)=\frac{x^3}{1+2x^2}$$ where both $x$ and $T$ are measured in units of $\mathrm{\$}10,000.$ It is given that $$T^{\prime }(x)=\frac{x^2(2x^2+3)}{(1+2x^2{)}^2},\text{and},T^{″}(x)=\frac{2x(3-2x^2)}{(1+2x^2{)}^3}.$$ The tax system is criticised as being unfair to middle-income earners. Give mathematical evidence for this claim.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Compare the marginal tax rate $T^{\prime }(1)$ with the long-run average tax rate $\underset{x\to \mathrm{\infty }}{\lim }\frac{T(x)}{x}$.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

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.)

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