### John Staddon, Ph.D.

# Bernoulli and the Taxman, Part I: Fair Tax

## Is the income tax fair?

Posted Apr 24, 2016

There are very few real laws in social science. One of the oldest was discovered independently in political science and in psychophysics (the study of sensation). Daniel Bernoulli (1700-1782), a brilliant Swiss mathematician and discoverer of the principle that allows airplanes to fly, also studied risk. He proposed that the way people value money or goods shows *diminishing returns*: a dollar added to wealth of $10 means much more than a dollar added to a base of $1000.

Being a mathematician, he turned this into a quantitative law: that value is logarithmically related to amount. In other words, 10% of my wealth means as much to me as 10% of your (much greater) wealth means to you. Utility is all about ratios.

This works for risk, because it means that each increment of wealth is valued less and less. Consequently, a 50% chance of winning $100, two increments of $50 each, will be worth less to you than 100% chance of $50 – because the second $50 has less utility than the first. Half of 50 plus something less than 50 is obviously less than 50. This is called risk aversion.

Some years after Bernoulli, two Germans, Ernst Weber (1795-1898) founder of experimental psychology, and Gustav Fechner (1801-1887) a physicist and philosopher, were responsible for a similar psychological principle. The Weber-Fechner law deals not with value but with our ability to make comparison judgments. Weber showed (for example) that if you can just tell the difference between a 10lb weight and an 11lb weight, you will find it equally hard to tell a 50lb weight from a 55lb. Again ratios win and the law is that the effect of a stimulus – brightness, loudness, etc. – is logarithmically related to its physical intensity. A series of sound intensities in decibels 100, 90, 80, etc. are in fact in the ratios 100, 10, 1, etc.: equal differences in decibels, a logarithmic unit that corresponds roughly to felt loudness, correspond to equal ratios of physical intensity. So perception, also, is all about ratios.

What does this have to do with income tax? Three things are important to setting a schedule of income tax rates: How much money will it make for the government? How will it affect the economy: will GDP grow or shrink? And, is it fair?

Economists are much more interested in the first two questions than in the last because the government’s total take and the effect of a tax schedule on the economy are interrelated. If a tax reduction increases economic growth, it may also increase total revenue despite the rate reduction. Conversely, if growth is impeded, an increased tax rate may actually yield less total revenue. There is little agreement about the real effect of income tax rates on economic growth.

But fairness is something where Bernoulli and Weber-Fechner do have something to say. If equal ratios are also equal utilities – are valued equally – then a fair tax is just a flat tax. Take the same fraction of everyone’s income and equal pain should be felt by all. Agreed?

Unfortunately, there are two things wrong with this attractively simple proposal, one philosophical and one practical. The philosophical one is that although Bernoulli’s theory works for an individual – or at least most individuals – it may not work for all. I may value my first dollar much more than my thousandth, as Bernoulli proposed, and that may also be true for my friend (unless he is a miser, who values every dollar the same). But that is not the same as saying that he values his first dollar exactly as much as I value mine. We can compare values for the same person, but we can’t be sure one person’s valuation is the same as another’s.

Nothing can be done about the epistemological objection to interpersonal comparison. But we can’t ignore the fact that Bernoulli’s law breaks down for the very poor. Losing a dollar from an income at subsistence level surely is a very high cost to a poor person. The simple flat tax fails this test. Taking the same faction of a very poor person’s income as of a very rich is obviously unfair because the cost to the very poor person is obviously greater.

One solution, adapted by most flat-tax proposals, is to have some kind of floor, an income below which no tax at all is paid.

The graph shows the actual fraction of income paid in tax for every level of income when the tax ‘floor’ is $10,000: zero tax below $10K and then 20% of each dollar thereafter. The tax-take fraction increases, rapidly at first, then slowing down as income increases. Finally at $300K or so it is close to the 20%, the flat-tax level.

I’ll show next time what’s wrong with this particular solution to the fair-tax problem.