You're thinking of IRS accounting wrongly.

I would assume (and always question assumptions) that tax revenues would be nonzero if IRS funding were zero. Some people will just voluntarily pay their taxes out of a sense of civic duty. Then, for the first dollar spent on the IRS, that will return some multiple of that dollar. I assume there are diminishing returns:

Revenue = integral( f(IRS_budget), 0, IRS_budget ) + C
f(1) > 4
f(current_IRS_budget + 1) = 4
f(current_IRS_budget + 2) < 4

That means that Congress decided to just stop funding the IRS at some point. In a sane business, this might be because all other projects return more than $4 for each additional dollar spent, and the revenues cannot immediately be used for additional spending. If there were no such restriction, all projects could be brainlessly funded to the point where only $0.99 in revenue is realized from an additional $1 in budget.

The head tax simply has a different function: Revenue = integral( g(IRS_budget), 0, IRS_budget ) + D

We don't know what the shape of that function might be, only that there is a hard upper limit on revenue equal to the capitation * population.

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You're confusing equality with fairness. A capitation would be equal. A fixed percentage after deduction would be fairer. I might argue that a different tax equation would be even fairer than that. But government is not, and never will be, fair. The minimum amount of fairness we should tolerate is equality.

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You are missing the point about incentives for extra work. An extra $10M is so far beyond the scenario I had in mind that you might as well be talking about trade routes in the Tau Ceti system.

In the real world, at the very bottom, payroll taxes are very significant, as they are a regressive tax. Means testing on subsidy programs also play a role. So some people face a choice such that if they work harder, they actually get less disposable income. If you get $200 (gross) a week from working 20-30 hours part time at the supermarket, and you get WIC and/or SNAP benefits, due to your poverty, you are still paying $12.40 plus unemployment tax, even with no income tax withholding, for $187.60 net. If you become full time, working 40 h/wk at $7.35/h, you lose your SNAP benefit. And you are now paying $18.20 in payroll taxes per week. You now have $275.45/wk to spend. If your SNAP benefit was more than $87.85, you are doing more work for less disposable income. The average SNAP benefit is $127/person.

Accounting wise, the loss of subsidy is roughly equivalent to an additional tax on the extra earned income. If the only tax/subsidy were a capitation, no one would ever refuse promotion from part time to full time and a raise from $7.25/hr to $7.35/hr out of fear that they would end up with less to spend. (Again, not necessarily a fair way to encourage additional work.)

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The government's ability to borrow and run the money-printing presses is orthogonal to discussion on forms of taxation.

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I am not advocating a head tax. I am simply stating possible arguments in its favor, playing the devil's advocate. It is pretty much the exact opposite of guaranteed basic income, which I think would be more favorable. But a capitation is not an unmitigated negative. It does have its good points, which have to be compared with other systems. You said you couldn't think of any positives about it, so I gave you some to consider. I am not extremely motivated to defend any of those points, but I hope you will concede that they do exist.