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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 $T(Y)/Y$.

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  • $\begingroup$ Is it assumed that there is no per capita tax, that is $T(0) = 0$? And is it assumed that $T(Y)$ is continuous? $\endgroup$ – Giskard Jun 4 at 14:48
  • $\begingroup$ 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. $\endgroup$ – Yejin Jun 5 at 4:24
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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.)

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    $\begingroup$ Actually, isn't this function regressive in terms of marginal tax rate, since the second derivative is negative? $\endgroup$ – Yejin Jun 6 at 4:35
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    $\begingroup$ Seems like you are right! Will edit later. $\endgroup$ – Giskard Jun 6 at 4:41
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    $\begingroup$ @Yejin I edited the answer. $\endgroup$ – Giskard Jun 6 at 12:35
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$$ \frac{\partial\frac{T(Y)}{Y}}{\partial Y}= \frac{T'(Y)}{Y} - \frac{T(Y)}{Y^2} $$

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.

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  • $\begingroup$ "you start with an average tax rate of zero" How so? $\endgroup$ – Giskard Jun 4 at 15:02
  • $\begingroup$ 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. $\endgroup$ – E. Sommer Jun 4 at 15:04
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    $\begingroup$ Why? Suppose $T(Y) = 20\% \cdot Y$. Then the average tax rate is constantly $20\%$, is it not? $\endgroup$ – Giskard Jun 4 at 15:33
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    $\begingroup$ 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. $\endgroup$ – Bill Clark Jun 4 at 22:47

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