# Pigovian Taxes and Income

[ See here for a more general and abstract (yet less mathy) proof. ]## Background

The standard economic response to a negative externality is a Pigovian tax Pigovian tax.

If a bee grower's bees sting the neighboring kids, then the government should impose a tax on beekeeping in residential areas, to discourage this.

However, I believe this analysis is oversimplified, because economists make the implicit assumption that a dollar tax to you is the same as a dollar tax to me.

This isn't true.

A minimal amount of self-reflection should reveal that the billionth dollar you make isn't as valuable to you as the thirteenth. Expert opinion and studies show that income has diminishing returns in making you happy Diminishing marginal utility of income? Caveat emptor.

Altering this assumption fundamentally changes how Pigovian taxes should be levied.

## A Simple Model

A simple way to model consumption is to theorize that Alice can divvy up her income to consume multiple goods. We will further assume that Alice's utility grows logarithmically with regards to each good's consumption.

From this, some calculus is enough to show that Alice will always spend the same amount on each good - regardless of how the price changes. To use the economic term, this implies that each good is unit elastic.

This allows us to model each good individually and compute Alice's utility as $$U = \ln{\frac{I}{P+T}}$$ where $I$ is the income Alice has allocated to the good, $P$ is the price of the good, and $T$ is the tax levied on the consumption of the good.

To consider Alice's contribution to social utility instead, we need to account for the revenue raised by a sales tax. To do this, we will define $R$ as the amount of utility society gets on the marginal tax revenue. This gives us a social utility function: $$U = \ln{\frac{I}{P+T}} + RT\frac{I}{P+T}$$

Solving this for the optimal tax just requires taking the derivative of $U$ with respect to $T$, setting that derivative to 0, and then solving for $T$. This yields $$T = IRP - P$$

The main takeaway from this is that our model implies the tax rate should increase with income - that is, it implies a very progressive tax. This makes sense, because our model isn't incorporating any changes in labor decisions.

This is a shortcoming of the model, but seeing as our primary purpose is to model Pigovian taxes, and these taxes are generally small as a proportion of income, this will probably have a minimal impact on our conclusions.

## Pigovian Taxes in the Model

So, let's add a negative externality to our model that scales linearly with consumption of a good. We'll let $x$ denote the size of the externality. Alice's contribution to social utility becomes $$U = \ln{\frac{I}{P+T}} + RT\frac{I}{P+T} - x\frac{I}{P+T}$$

Using calculus to solve again yields $$T = IRP - P + Ix$$

Note, that $IRP - P$ is the same as the tax as we got above (for a good with no externalities). The part of the tax that internalizes the externality is the $Ix$ term. This implies the sales tax should be increased in proportion to the size of both the externality and also Alice's income.

## Generalizing

Due to results like the fundamental theorems of welfare economics and the Atkinson-Stiglitz Theorem, the default response of most economists to calls for progressive Pigovian taxes is that they are irrelevant. The default belief is that policy should only maximize efficiency rather than equality with one exception: a progressive income taxation/transfer system. For this reason, for the above reason to be accepted as an interesting argument, we have to show that even if we have an optimal income tax/transfer system, progressive Pigovian taxes are *still* desirable.

Another way we should generalize this is to show the same results hold for goods that aren't unit-elastic.

Unfortunately, generalizing in these two ways makes the math significantly more complicated. I actually kind of want to try to publish the results, but I'll merely put a sketch of the proof here. For simplicity, I'll use carbon emission as the thing to tax. Alright, here goes nothing...

Suppose (1) the externality from CO2 is small relative to total income and (2) we live in a society with an optimal income tax system.

Now, the general worry with any kind of Pigovian tax is that it will distort labor by altering the marginal and effective tax rates at different income levels. For instance, suppose rich people spend a smaller portion of their income on electricity. This implies that the Pigovian tax would fall as a percentage as income increases - that is, the Pigovian tax would raise effective marginal income tax rates.

To get around this problem, we will give tax credits to people equal to the Pigovian tax applied to the average CO2 caused by people at their income level. For instance, if the average person making \$80k emits 2000 pounds of CO2 per year and the Pigovian tax is \$2 per pound, we'll give everyone making \$80k \$4000 in tax credit. In this way, we ensure the average marginal and effective tax rates at each income level remain the same. In plain English, this means, the tax system is no more/less progressive after the Pigovian tax is implemented.

However, beyond the average, people will face different marginal and effective tax rates depending on their CO2-income elasticity and CO2 consumption. However, if we assume the effects of taxes on labor is locally a line (i.e. the slope is non-zero and the second derivative is reasonably small), then the behavioral change caused by some people facing slightly effective higher tax rates will be perfectly offset by people at the same income level facing effectively slightly lower tax rates. The overall effect on labor will be 0, and this strongly suggests that this means the original optimal income tax system will remain optimal after we add (a) the Pigovian tax and (b) the tax credit discussed above.

Now we reach the final step. Nothing in the above analysis assumes the Pigovian tax is a fixed number. In fact, we can make it any function of income we want and the above results hold so long as its size is small compared to a person's income.

Typically, we set the Pigovian tax equal to the surplus lost by the externality. However, there what we actually care about is the social welfare (utility) lost, not surplus. So, in order to properly internalize the tax, we need to convert from surplus-lost to utility-lost to induce the ideal change among consumers. If $u(x)$ yields the utility someone gets from income, then this conversion factor is just $u'(x)$.

In particular, if $u(x) = \log(x)$, then $u'(x) = 1/x$, which implies the Pigovian tax should be proportional to income. If, on the other hand, $u(x) = -x^{-0.35}$ (my preferred model), then the Pigovian tax should be proportional to $x^{1.35}$.

## Conclusions and Caveats

Now, I must admit there are large practical and political concerns with implementing a system based on the above analysis, but I think the claim that Pigovian taxes should be *roughly* proportional to income seems reasonable. In fact, at least one country levies traffic fines in proportion to your income, and since fines are effectively Pigovian taxes, this is a real-world example of the such an implementation (though probably with less mathematical justifiaction)

Morever, there are still some practical applications. For instance, since smokers tend to earn less than non-smokers The Economic Consequences of Being a Smoker. he above anslysis suggests that standard estimates of the optimal cigarette tax will be higher than the true optimum.

Likewise, all the above applies equally well to taxing "sin taxes" that help internalize internalities.

*Wikipedia, The Free Encyclopedia*. Retrieved 21:46, November 9, 2019, from https://en.wikipedia.org/w/index.php?title=Pigovian_tax&oldid=925315154

*Social Indicators Research, 70*(3), 243-255. https://doi.org/10.1007/s11205-004-8393-4