John Staddon, Ph.D.

Adaptive Behavior

Bernoulli and the Taxman, Part II: Fair (and Painless)?

Why is income tax so confusing?

Posted May 09, 2016

Bernoulli’s principle of diminishing marginal utility says that the felt cost–the pain–of a cut in income is proportional to income level.   Losing $1000 out of an income of $20,000 a year causes the same amount of pain as coughing up $10,000 from an income of $200,000.  This is the moral basis for a flat income tax, a tax which takes the same income proportion (but more actual cash) from the high-paid than the low.  (A progressive tax, on the other hand takes a disproportionate amount of income from the higher-paid.  More on progressive tax in a moment.)

I showed in Part I that this plausible scheme has at least one serious flaw: it doesn’t take account of the obvious fact that the pain of any loss gets very high when income is close to subsistence level.  If you’re barely making it, losing even five or ten dollars may be catastrophic.  Bernoulli’s principle doesn’t apply when you are close to starving. 

The simple solution to this problem is a tax floor: a threshold income level below which no tax is paid.  The result is a curve like this which shows the actual income fraction paid as tax for different income levels.  With a “tax floor” the result is a higher fraction is paid as income increases.  Only high incomes pay close to the actual flat-tax rate.  Flat-tax-with-a-threshold is weakly progressive.

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Source: JS image

This kind of graph–percent income paid at each income level–is the best and simplest way to look at any income-tax plan.

So does adding a tax floor solve the problem of fairness to both high and low incomes?  Yes and no:  “yes” it saves the very poorest from having to pay tax.  But as soon as tax kicks in, above $10K in the picture, it hits at the maximum rate–hence the steep rise in net tax rate after $10K. 

Another psychological principle is this: people pay much more attention to sharp changes than to gradual ones.  This effect shows up in (for example) visual illusions, where you may see a series of increasingly dark steps when the visual image consists of a series of slow actual brightness increases followed by sharp declines, as in the picture.  Each step is in fact the same average brightness, but you see a descending series because the visual system is especially sensitive to the sharp declines.

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Source: JS image

This edge effect as it is called means that the sudden increase in  marginal tax rate as a person’s income rises above the tax ‘floor’ will be very noticeable.  People may hesitate to cross the threshold because the first over-threshold dollar bite will have the biggest effect–because the felt value, the extra dollar divided by the base amount (the Weber fraction), will be smallest for that first dollar.  Many workers who might otherwise do overtime or a second job will therefore hesitate to cross the tax threshold.  Because of Weber’s law, that first increment will  actually be less attractive than later ones, which is surely the opposite of what is needed to encourage people to do more. 

“Fine” you may say.  “Why should we encourage people to work harder?”  Maybe, but surely we should not discourage extra effort, which is what the flat-tax threshold actually does?

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Source: JS image

The next picture shows an alternative.   The flat tax with a threshold is defined by two numbers: the threshold and the flat rate.  The alternative in the picture, called a logistic curve, is also defined by two numbers: the maximum tax rate and the rate at which the maximum is approached.  The picture shows two examples: gradual, the solid blue line increases very slowly from a low base level only reaching 20% at an income of more than $220,000 or so.  The maximum is at 50% which is reached at an income of  more than $300,000.  The dashed rapid curve gets to the 20% level by an income of $100,000 and reaches the maximum tax rate by an income of only $150,000 . 

Both curves introduce the increased rate gradually, avoiding the deterrent effect of the sudden increase built in to the flat-tax-with-threshold. 

What does the actual US tax structure look like?  Forget for the moment that tax rates are just the cover illustration on a book of 70,000+ pages of exemptions, exceptions and credits.  Published tax rates are just the tip of an iceberg.  Still they should make some sense.  Do they?

Wikimedia commons
Source: Wikimedia commons

The existing structure involves seven numbers–rates from 10 to 39.6%–which makes it hard to grasp.  But we can plot it in the same net-tax form as the schemes I’ve been discussing . The blue curve in the picture (from 2013) shows what the current tax scheme looks like   It looks more like the flat tax than the logistic.  It’s also much more generous to high incomes than my examples.  The maximum, 39.6% ceiling isn’t reached even by $1,000,000 earners. 

The bottom line:  Instead of getting bogged down in the details of seven or more tax rates, let’s just take the logistic scheme as a starting point.  Can we balance fairness, efficiency and total take and agree on at least those two numbers: the maximum rate, and how rapidly it is reached?