For years I’ve held two beliefs fairly confidently:

- We should not have a flat tax (in which everyone’s income is taxed at a constant percentage) because the relationship between losing some percent of one’s income and the reduction in one’s happiness changes depending on income level.

Our taxes should vary by income, so that the experienced burden of maintaining the country is felt equally among all taxpayers.

Different incomes should pay different tax rates so that we each feel our taxes equally. - The effect that income has on happiness is logarithmic, following the form:

h(x) = a+b Log(x)

where x is income and h(x) is happiness as a function of income.There is actually good evidence for this, as can be seen in the plot below.

A University of Michigan study showed this relationship more or less holds for each of the 13 countries surveyed.For the US in particular, we can refer to the Gravity and Levity blog for this very nice fit:

which holds for the range of data collected (incomes from 15k to 115k per year).

**The problem is that these two beliefs actually are completely contradictory.
**

Lets use the following convention:

x: Income per year

h(x): Happiness resulting from that income

t(x): The tax paid for each income

Δ: The uniform happiness reduction we want our taxes to enact

Suppose h(x) = a+b Log(x). What kind of tax should we have to make sure that regardless of income, x, everyone’s happiness is reduced equally?

(i.e. We all ‘feel’ taxes equally, regardless of income)

We can determine t(x) as follows:

Δ = h(x) – h(x-t(x))

Δ = (a+b Log(x)) – (a+b Log(x – t(x)))

-Δ/b = Log(1-t(x)/x)

t(x) = x (1-Exp[-Δ/b])

In other words, a constant fraction, (1-Exp[-Δ/b]), of everyone’s wealth is taken.

This is a flat tax.

**A few notes about this:**

First of all, the Michigan data only covers an annual income of 15k to 115k. Clearly the a + b Log(x) trend can’t extend indefinitely — happiness ratings only go up to 10 — so at some point it must level off.

In both limits, it becomes unclear how to treat taxes. Inevitably the tax code would need to become progressive, making absurdly wealthy people (who, on average, can’t theoretically get much happier) pay huge percentages.

I’m also sure we wouldn’t want a society in which people are taxed into misery — one insisting that taxes are taken even from people who are on average at 1/10 happiness. Figuring out how the happiness vs income curve should behave in the low limit, and what to do with that information, is something we would have to think about.

**What’s the point**

In the end, this short calculation doesn’t really say anything about how we ‘should’ tax at all.

If you believe that a completely progressive tax is best, you can still think that. You don’t have to believe either of the premises, the data I show, or or even the simple arithmetic was done.

What’s important (to me), and the reason that I’m writing any of this down, is that from the start I believed both of the premises, and I would have agreed with the math I did to compare them. Yet I had no idea that they were in conflict with each other.

Making things worse is that as beliefs go, these ones are on the quantitative, more verifiable end. We all have an unlimited supply of beliefs that are infinitely more vague than “well-being follows a logarithmic relationship with income”.

The Boolean Satisfiability Problem (also called ‘SAT’) involves checking the truth values of a set of variables connected by boolean operations (AND, OR, NOT).

For example, suppose I hold n beliefs (each of which can be either true or false) all connected by AND, OR, and NOT.

(belief_1 AND belief_2 AND belief_3) OR (belief_3 AND belief_1) OR … (belief_n AND NOT(belief_1) AND belief_(n-1))

The SAT problem is to determine the values of the variables (the beliefs) which make the entire statement true.

Checking the truth value of a long boolean statement composed of True, False, AND, OR, and NOT actually sounds pretty simple. In particular, it sounds infinitely simpler than checking whether all of the uncountable and potentially nebulous beliefs of a single human being agree with one another.

Unfortunately, SAT happens to be the first problem in computer science to ever be proven NP-complete. In its worst case, the computational complexity of the problem makes it untenable. It is the low-hanging fruit of incredibly hard problems.

We’re all inevitably walking around in a mess of cognitive dissonance, with some huge set of our beliefs in contradiction with one another.

With all that in mind, we should probably all be dramatically less confident in what we think is true.