<p>In comments it was clarified that <span class="math-container">$T(Y) = 0$</span> and continuity are not assumed. In this case there are several counterexamples, a relatively simple one being
<span class="math-container">$$
T(Y) = \left\{
\begin{array}{ll} 
1 + 20\%Y &amp; \text{if } Y \leq 5 \\
40\% Y &amp; \text{if } Y &gt; 5.
\end{array}
\right. 
$$</span>
Here the taxation is progressive, but the average tax rate <span class="math-container">$T(Y)/Y$</span> is close to infinite near <span class="math-container">$Y = 0$</span> as
<span class="math-container">$$
\lim_{Y \to 0} \frac{1}{Y} + 20\% = \infty,
$$</span>
and <span class="math-container">$T(Y)/Y$</span> is smaller later, e.g. <span class="math-container">$T(5)/5 = 2/5$</span>.</p>

<p>As long as <span class="math-container">$T(0) &gt; 0$</span> or there is a positive jump at any income level <span class="math-container">$Y$</span>, the average tax rate may not be monotonically increasing.</p>

<p>(The reader may use the math of the cost functions <span class="math-container">$AC,MC$</span> to examine this.)</p>

CC BY-SA 4.0
2019-06-06T04:35:16.230Z
Actually, isn't this function regressive in terms of marginal tax rate, since the second derivative is negative?
Yejin
CC BY-SA 4.0
2019-06-06T04:41:06.577Z
Seems like you are right! Will edit later.
Giskard
CC BY-SA 4.0
2019-06-06T12:35:34.853Z
@Yejin I edited the answer.
Giskard
CC BY-SA 4.0
2019-06-05T05:52:59.543Z
2019-06-06T12:34:45.147Z
2019-06-06T12:34:45.147Z
Giskard
<p><span class="math-container">$$
\frac{\partial\frac{T(Y)}{Y}}{\partial Y}= \frac{T'(Y)}{Y} - \frac{T(Y)}{Y^2}
$$</span> </p>

<p>This can only be smaller than 0 if </p>

<p><span class="math-container">$$ T'(Y) &lt; \frac{T(Y)}{Y} $$</span></p>

<p>In other words, the marginal tax rate needs to be smaller than the average tax rate. This can't happen because you start with an average tax rate of zero and a positive marginal tax rate. From then on, in a progressive scheme, the tax due increases faster than taxable income. Progressive taxation is actually characterized by the fact that the marginal tax rate exceeds the average one. <a href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=931128" rel="nofollow noreferrer">See here, p.27</a>.</p>

CC BY-SA 4.0
2019-06-04T15:02:19.603Z
"you start with an average tax rate of zero" How so?
Giskard
CC BY-SA 4.0
2019-06-04T15:04:58.493Z
if your income is zero, you don't pay taxes. at some point, you will start to pay one \$ on some amount of income. So your average tax rate will be very close to zero when taxation kicks in.
E. Sommer
CC BY-SA 4.0
2019-06-04T15:33:25.997Z
Why? Suppose $T(Y) = 20\% \cdot Y$. Then the average tax rate is constantly $20\%$, is it not?
Giskard
CC BY-SA 4.0
2019-06-04T22:47:34.593Z
Giskard is correct that the fact that $T(0)=0$ tells us nothing about the _rate_ at 0 income (and an argument involving limits can get around the 0/0 problem for computing the average) but I think it doesn’t actually matter.  I think the rest of the argument holds anyway.
Bill Clark
CC BY-SA 4.0
2019-06-04T14:56:57.167Z
2019-06-04T15:03:18.673Z
2019-06-04T15:03:18.673Z
E. Sommer
<p>Or does progressive marginal taxation imply that it is a progressive average taxation?</p>

<p>Here, I mean that if <span class="math-container">$T(Y)$</span> is a tax function of income, marginal tax rate would be <span class="math-container">$dT/dY$</span> and average tax rate would be <span class="math-container">$T(Y)/Y$</span>.</p>

CC BY-SA 4.0
2019-06-04T14:48:44.643Z
Is it assumed that there is no per capita tax, that is $T(0) = 0$? And is it assumed that $T(Y)$ is continuous?
Giskard
CC BY-SA 4.0
2019-06-05T04:24:16.857Z
No none of these are assumed. It just needs to be well defined. But then we will have to talk about intervals. For example, the function $T(Y)=0$ for $0<Y<10$, and $T(Y)=Y-10, 10<Y$ is progressive in marginal tax if we view it as intervals, but constant if we consider the progressivity within the intervals.
Yejin
<taxation><public-economics>
Can a progressive marginal taxation be regressive in terms of average tax rate?
Yejin
# Can a progressive marginal taxation be regressive in terms of average tax rate?

(Asked by: username_0 on 2019-06-04)

Or does progressive marginal taxation imply that it is a progressive average taxation?

Here, I mean that if $T(Y)$ is a tax function of income, marginal tax rate would be $dT/dY$ and average tax rate would be $T(Y)/Y$.

**Tags:** taxation, public-economics
> **Comments:**
> **username_1**: Is it assumed that there is no per capita tax, that is $T(0) = 0$? And is it assumed that $T(Y)$ is continuous?  
> **username_0**: No none of these are assumed. It just needs to be well defined. But then we will have to talk about intervals. For example, the function $T(Y)=0$ for $0<Y<10$, and $T(Y)=Y-10, 10<Y$ is progressive in marginal tax if we view it as intervals, but constant if we consider the progressivity within the intervals.  

---

## 2 answers

### Answer by username_1 on 2019-06-05. Score: 1 | Accepted answer

In comments it was clarified that $T(Y) = 0$ and continuity are not assumed. In this case there are several counterexamples, a relatively simple one being
$$
T(Y) = \left\{
\begin{array}{ll}
1 + 20\%Y & \text{if } Y \leq 5 \\
40\% Y & \text{if } Y > 5.
\end{array}
\right.
$$
Here the taxation is progressive, but the average tax rate $T(Y)/Y$ is close to infinite near $Y = 0$ as
$$
\lim\_{Y \to 0} \frac{1}{Y} + 20\% = \infty,
$$
and $T(Y)/Y$ is smaller later, e.g. $T(5)/5 = 2/5$.

As long as $T(0) > 0$ or there is a positive jump at any income level $Y$, the average tax rate may not be monotonically increasing.

(The reader may use the math of the cost functions $AC,MC$ to examine this.)

> **Comments:**
> **username_0**: Actually, isn't this function regressive in terms of marginal tax rate, since the second derivative is negative?  
> **username_1**: Seems like you are right! Will edit later.  
> **username_1**: @Yejin I edited the answer.  

---

### Answer by username_2 on 2019-06-04. Score: 3

$$
\frac{\partial\frac{T(Y)}{Y}}{\partial Y}= \frac{T'(Y)}{Y} - \frac{T(Y)}{Y^2}
$$

This can only be smaller than 0 if

$$ T'(Y) < \frac{T(Y)}{Y} $$

In other words, the marginal tax rate needs to be smaller than the average tax rate. This can't happen because you start with an average tax rate of zero and a positive marginal tax rate. From then on, in a progressive scheme, the tax due increases faster than taxable income. Progressive taxation is actually characterized by the fact that the marginal tax rate exceeds the average one. [See here, p.27](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=931128).

> **Comments:**
> **username_1**: "you start with an average tax rate of zero" How so?  
> **username_2**: if your income is zero, you don't pay taxes. at some point, you will start to pay one \$ on some amount of income. So your average tax rate will be very close to zero when taxation kicks in.  
> **username_1**: Why? Suppose $T(Y) = 20\% \cdot Y$. Then the average tax rate is constantly $20\%$, is it not?  
> **username_3**: Giskard is correct that the fact that $T(0)=0$ tells us nothing about the _rate_ at 0 income (and an argument involving limits can get around the 0/0 problem for computing the average) but I think it doesn’t actually matter.  I think the rest of the argument holds anyway.  

---

