imagine you could choose between the two following tax formulas.

1. `10% * income — 4000`

if you make 40,000\$ salary, both formulas are equivalent: you pay zero in tax. if you make more, then the first formula is preferable. if you make less, the second formula is preferable.

let me restate that clearly: the poor will prefer the higher tax rate.

for instance, say you make 39,000\$. then your tax “bill” under these two formulas is:

1. `10% * 39,000 — 4000 = -100` (i.e. a 100\$ refund)

the negative numbers mean you get a refund. this is called a negative income tax (NIT), and is effectively the same thing as a universal basic income (UBI). clearly someone making below 40,000\$ prefers the higher 20% tax rate to the lower 10% tax rate.

but suppose you make 41,000\$. then the amounts are:

1. `10% * 41,000 — 4000 = 100`

these are positive values, meaning you pay taxes rather than getting a refund. someone making above 40,000\$ prefers the lower 10% tax rate to the higher 20% tax rate.

# the poor prefer higher interest rates

we just saw that a 20% tax rate is preferable for poorer people and undesirable for wealthier people, compared to the lower 10% tax rate. this directly contradicts progressive taxation dogma, which says that it’s better for the poor to pay lower tax rates.

as we see from these simple examples, there’s more to your tax bill than merely the percentage rate you pay. by offsetting that percentage with a fixed size “negative income tax”, we can produce a tax system which is progressive in the aggregate, even tho it uses flat percentages.

# graphing it

i made a quick approximation of the effect of a flat tax vs progressive tax, with all revenues in excess of government expenditures going back evenly to all citizens in the form of the aforementioned negative income tax.

blue is the real net worth percentiles from the USA, which gives us a realistic sense of inequality regardless of whether we’re speaking in terms of wealth or income. the red line is the new distribution if we enact a 40% flat tax. the yellow line is where we end up with a progressive tax using a formula of `40% / (1 + (100,000 / net_worth))`; this starts at a 0% rate and gradually increases to 40% at the top percentile.

what jumps out here is that the flat tax (red) is better than the progressive tax (yellow) for everyone under the 48th wealth percentile! that is, the flat tax is more “progressive” than the progressive tax!!! (note that the red and yellow lines almost perfectly overlap, so you’ll want to click on the image to zoom in to get a clear look.)

## simplified

imagine we have 3 people making 0, 200,000, and 1,000,000 per year. with the flat tax at a maximum 100% rate, they each net 400,000 (1.2M/3).

with a progressive tax, the total revenue is 0 + 40,000 + 1,000,000 = 1,040,000, so the nets are 346,667, 426666, and 346,667. so at a certain point, you have a disincentive to work.

# practicality

furthermore, all this is to say nothing of the practical logistical concerns of implementing progressive taxation. for instance, finding all wealth (outsmarting tax avoidance) and objectively assessing assets in order to determine net worth. normal “transaction” taxes, such as income tax, sales tax, and value added tax (VAT) are convenient because they come with an objective market price to which we can simply slap a flat percentage tax. wealthier people will pay more of these taxes in total, merely by virtue of the fact that they consume more goods and services. these types of taxes are far cheaper to administer.

and what about illiquid assets? how much do we tax illiquid holdings in a pre-money venture startup? and how to we extract the money given there’s not yet a market for those assets?

and if we attempt a progressive income tax based purely on income and not net worth, that defeats the entire utilitarian point of progressivity in the first place. total wealth—not merely income—is what has decreasing marginal utility.

# conclusion

progressive taxation advocates have their hearts in the right place. but they’re stuck on a basic error of arithmetic, thinking only in terms of the percentage rate, while ignoring the power of a negative constant in the formula.

thanks to @wvaughn3 for suggesting a better progressive tax formula than the exponential one I initially employed.

this piece focuses on pure math. i highly suggest this much deeper dive if you’re looking for more empirical comparative international data about the long-term effects of progressivity.

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