# 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.

## Conclusions and Caveats

Now, while I believe that modeling utility as the logarithm of consumption isn't too far from the truth, it is obviously a simplification. That being said, I think the claim that Pigovian taxes should be *roughly* proportional to income seems reasonable. Even if you don't believe this, you should accept that Pigovian taxes should increase with income - at least in theory.

Alas, in practice it is probably too economically, socially, and politically costly to customize tax rates to individual people. Still, it's nice to at least have a theoretically correct policy.

However, there is still some practical applicability. For instance, since smokers tend to earn less than non-smokers The economic consequences of being a smoker, it suggests that standard estimates of the optimal cigarette tax will be higher than the true optimum.

It's also worth noting that the exact same argument applies to fines: they, too, should be proportional to income. To prove this, you just multiply the "tax" by the probabilty of being caught.

*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.