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	economics inequality Pigovian Taxes II Introduction In any introductory microeconomics, you'll learn that externalities should be addressed by Pigovian tax equal to the size of the externality. The fundamental problem with this analysis, like much of microeconomics, is that it deliberately ignores distributive effects. The common defense is that we should aim to maximize surplus and then solve distributive issues after the fact. Frequently, defenders will often mumble something about the efficiency of lump sum transfers even though this is ludicrous in the real world. The more cogent defenders will instead mumble something about the Atkinson-Stiglitz theorem. However, what follows is a proof sketch I've come up with showing that even with idealized income taxation that is optimal from an efficiency-equality perspective, a utilitarian framework still implies significant adjustment to naive Pigovian taxes: namely that people with higher incomes should pay higher Pigovian taxes. If this sounds familiar, you may have read my earlier work on the subject, which was both more mathy and less general. Alright, without further ado, here is the sketch. Proof Sketch I'll make three assumptions: There exists some know function $u$ that maps income to utility. The externality we're talking about is small with respect to total consumption. We already have optimal income taxes and transfers in place. The last assumption is important because it allows us to avoid what I mentioned earlier: having our distributive concerns be rejected because of the second fundamental theorem of welfare economics and the Atkinson-Stiglitz theorem. Alright, for concreteness, let's talk about the externality associated with driving a heavier car: that you increase the mortality risk of other drivers. First note that people of different incomes will spend different proportions of their budget on car weight, which means adding even a naive Pigovian tax effectively changes our marginal and effective income tax rates at each point on the distribution. Since our income tax is already optimal, that's something we want to avoid. For this reason, we will let each consumer deduct the average weight of cars purchased by other consumers with the same income. By doing this, the average marginal and average effective tax on income stays the same at all income levels. Since we're assuming this Pigovian tax is small, we can ignore second order-effects, which means lets us ignore the heterogeneous effects of this Pigovian tax. However, the exact same logic lets us add any (small) Pigovian tax without affecting the optimality of our income tax system. This lets us tweak it to optimize utility rather than surplus without worrying that we're mucking up the efficiency-equality tradeoff or, more concretely, changing the overall labor supply at each wage level. In other words, now we can just optimize social welfare (utility) within the car market and ignore effects on the labor market. So what Pigovian tax optimizes utility in the car market? Well, I've addressed the logarithmic case before, but, in general, we just need to take the utility lost to car weight and convert it into dollars so that each person accounts for it properly. We can do this by just taking the utility lost and dividing by $u'$, which is, by definition how much utility each dollar gives. If, for instance, $u(x)=\ln{x}$, then $u'(x)=1/x$, which means the Pigovian tax should be proportional to your income. If, on the other hand, $u(x)=-x^{-0.35}$ like I believe, the Pigovian tax should be proportional to your income raised to the 1.35 power. Finally, note that every part of this proof applies equally well to Pigovian taxes on internalities. Real World Considerations In the real world, there are some additional things to consider: Pigovian taxes might not always be small. If so, they also distort the labor market, which makes the above yield numbers that are too large. I haven't thought a lot about this, but it probably also means the conclusions above would be a little too progressive. Instead of letting people deduct the average consumption for their income, you could also equivalently just tweak the income tax system to account for this deduction. After this tweaking, we can eliminate the deduction. Obviously, these adjustments are politically infeasible and fairly invasive of people's privacy. You would have to, for instance, provide your income even if you bought your car with cash.

economics inequality

Pigovian Taxes II

Introduction

In any introductory microeconomics, you'll learn that externalities should be addressed by Pigovian tax equal to the size of the externality. The fundamental problem with this analysis, like much of microeconomics, is that it deliberately ignores distributive effects.

The common defense is that we should aim to maximize surplus and then solve distributive issues after the fact. Frequently, defenders will often mumble something about the efficiency of lump sum transfers even though this is ludicrous in the real world. The more cogent defenders will instead mumble something about the Atkinson-Stiglitz theorem.

However, what follows is a proof sketch I've come up with showing that even with idealized income taxation that is optimal from an efficiency-equality perspective, a utilitarian framework still implies significant adjustment to naive Pigovian taxes: namely that people with higher incomes should pay higher Pigovian taxes.

If this sounds familiar, you may have read my earlier work on the subject, which was both more mathy and less general.

Alright, without further ado, here is the sketch.

Proof Sketch

I'll make three assumptions:

There exists some know function $u$ that maps income to utility.
The externality we're talking about is small with respect to total consumption.
We already have optimal income taxes and transfers in place.

The last assumption is important because it allows us to avoid what I mentioned earlier: having our distributive concerns be rejected because of the second fundamental theorem of welfare economics and the Atkinson-Stiglitz theorem.

Alright, for concreteness, let's talk about the externality associated with driving a heavier car: that you increase the mortality risk of other drivers.

First note that people of different incomes will spend different proportions of their budget on car weight, which means adding even a naive Pigovian tax effectively changes our marginal and effective income tax rates at each point on the distribution. Since our income tax is already optimal, that's something we want to avoid.

For this reason, we will let each consumer deduct the average weight of cars purchased by other consumers with the same income. By doing this, the average marginal and average effective tax on income stays the same at all income levels. Since we're assuming this Pigovian tax is small, we can ignore second order-effects, which means lets us ignore the heterogeneous effects of this Pigovian tax.

However, the exact same logic lets us add any (small) Pigovian tax without affecting the optimality of our income tax system. This lets us tweak it to optimize utility rather than surplus without worrying that we're mucking up the efficiency-equality tradeoff or, more concretely, changing the overall labor supply at each wage level. In other words, now we can just optimize social welfare (utility) within the car market and ignore effects on the labor market.

So what Pigovian tax optimizes utility in the car market? Well, I've addressed the logarithmic case before, but, in general, we just need to take the utility lost to car weight and convert it into dollars so that each person accounts for it properly. We can do this by just taking the utility lost and dividing by $u'$, which is, by definition how much utility each dollar gives.

If, for instance, $u(x)=\ln{x}$, then $u'(x)=1/x$, which means the Pigovian tax should be proportional to your income. If, on the other hand, $u(x)=-x^{-0.35}$ like I believe, the Pigovian tax should be proportional to your income raised to the 1.35 power.

Finally, note that every part of this proof applies equally well to Pigovian taxes on internalities.

Real World Considerations

In the real world, there are some additional things to consider:

Pigovian taxes might not always be small. If so, they also distort the labor market, which makes the above yield numbers that are too large. I haven't thought a lot about this, but it probably also means the conclusions above would be a little too progressive.
Instead of letting people deduct the average consumption for their income, you could also equivalently just tweak the income tax system to account for this deduction. After this tweaking, we can eliminate the deduction.
Obviously, these adjustments are politically infeasible and fairly invasive of people's privacy. You would have to, for instance, provide your income even if you bought your car with cash.