Can a progressive marginal taxation be regressive in terms of average tax rate?

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 $$T(Y)/Y$$.

• 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 Jun 4 '19 at 14:48
• 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 Jun 5 '19 at 4:24

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.)

• Actually, isn't this function regressive in terms of marginal tax rate, since the second derivative is negative? – Yejin Jun 6 '19 at 4:35
• Seems like you are right! Will edit later. – Giskard Jun 6 '19 at 4:41
• @Yejin I edited the answer. – Giskard Jun 6 '19 at 12:35

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

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.

• "you start with an average tax rate of zero" How so? – Giskard Jun 4 '19 at 15:02
• 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 Jun 4 '19 at 15:04 • Why? Suppose$T(Y) = 20\% \cdot Y$. Then the average tax rate is constantly$20\%$, is it not? – Giskard Jun 4 '19 at 15:33 • 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 Jun 4 '19 at 22:47