Question

An income tax system is considered 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

Hint:

Use ATR \(=T/Y\). A negative intercept (\(p<0\)) acts like a lump-sum transfer plus a proportional tax, making ATR climb with income.

Answer 1

The Correct Option is B

Step 1: Average tax rate (ATR).
\[ \text{ATR}(Y)=\frac{T}{Y}=\frac{p+tY}{Y}=\frac{p}{Y}+t \] Step 2: Progressivity condition (ATR rising with $Y$).
\[ \frac{d\,\text{ATR}}{dY}=\frac{d}{dY}\!\left(\frac{p}{Y}+t\right)=-\frac{p}{Y^{2}} \] For ATR to {increase} with $Y$: \(-\frac{p}{Y^{2}}>0 \Rightarrow -p>0 \Rightarrow p<0\).
Step 3: Check other signs.
If \(p>0\), ATR falls with \(Y\) (regressive). If \(p=0\), ATR is constant (= \(t\); proportional). Only \(p<0\) makes ATR rise with \(Y\) (progressive).
\[ \boxed{p<0\ \text{(must be negative)}} \]