2 marks, −0.667Public EconomicsTaxation: incidence and efficiencyProgressive, proportional and regressive taxesModerateabout 2 min

GATE 2023 Question 49: Taxation: incidence and efficiency

An income tax system is considered as progressive if the average tax rate rises with income. Consider an income tax schedule: $T = p + tY$ , where $T$ denotes the tax liability, $p$ is a constant, $t$ is the constant marginal tax rate, and $Y$ is the income. For this tax schedule to be progressive, the value of $p$

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Answer and worked solution

Answer: (B)

Average tax rate: $ATR = \dfrac{T}{Y} = \dfrac{p+tY}{Y} = \dfrac{p}{Y} + t$ .

For the tax system to be progressive, $ATR$ must increase as $Y$ increases, i.e. $\dfrac{d(ATR)}{dY} > 0$ .

$\frac{d(ATR)}{dY} = -\frac{p}{Y^2}$

For this derivative to be positive (ATR rising with income), we need $-p > 0$ , i.e. $p < 0$ .

Intuitively: if $p$ is negative, it acts like a fixed tax credit/rebate (everyone owes $tY$ but gets $|p|$ back), which matters proportionally more for low-income earners (a bigger share of their income) than high-income earners — so their effective average rate is lower, and the average rate rises as income rises. This is exactly what defines progressivity here.

Therefore the answer is (B) $p$ must be negative.

The trap: (A) must be positive

Students sometimes reason intuitively that 'a positive constant p, like a fixed tax amount everyone pays, sounds like it would make the system more burdensome on the poor and thus progressive' — but the actual math shows the opposite: a positive $p$ makes the tax regressive (ATR falls with income), while it's the negative p (a rebate/credit) that produces progressivity, which is the less intuitive direction without working through the calculus.

Plain-text version

An income tax system is considered as progressive if the average tax rate rises with income. Consider an income tax schedule: T = p + tY, where T denotes the tax liability, p is a constant, t is the constant marginal tax rate, and Y is the income. For this tax schedule to be progressive, the value of p (A) must be positive (B) must be negative (C) must be zero (D) can be any value except zero

Taxation: incidence and efficiency has come up 79 times in 7 exams since 2009.

What this question tests

This question is on Progressive, proportional and regressive taxes, part of Taxation: incidence and efficiency in Public Economics.

How the burden of a tax changes with income, measured by average and marginal tax rates.

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Question from the GATE 2023 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.