Yeah. It's a bit simple - like, we're not going down to infinite negative income tax for people below poverty, just assume zero tax paid for them.

To find C, you have to sum up the log10(R/P) for everyone (including corporations) in the nation and divide that into what the government needs. There's various ways to calculate what the government needs, perhaps you separate FICA and payroll tax income entirely if you keep that, or perhaps you just look at entire Federal spending, I believe around 3.8 trillion.

But to sum up the log10(R/P), you need income distribution data. Someone else did that very roughly with a spreadsheet, I haven't tried it myself. A few years ago when they did that, they figured 3.7 trillion for spending, and C was something like 8.86.

C would also be a bit higher if you were also trying to pay down the debt over a period of time.

Unfortunately, I don't understand the logic behind this. I would have thought that a flat tax in terms of utility would solve for the amount of taxes T in log10(R/P) - log10((R-T)/P) = C for some constant C across the population. That would correspond to taxing everyone an equal amount of utility, assuming a logarithmic model of utility.

If the tax itself is proportional to the utility function, that has the effect of taking much less utility from people with greater R. I'm also unclear on how your C works out. The largest company in the world makes less than $1 trillion per year, so even if P is $1, log10(R/P) is bounded above by 12. Given a US budget of several trillion USD, that seems to make C multiple orders of magnitude bigger.

Sorry if I'm missing something. That was my original reason for asking for a formula above.