Phil,
I was attending a political seminar tonight and got caught up in my own thoughts about this constitution I'm working on, and realized that you had pointed out a weakness in my system. I realized that I had also sort of caught it but just overlooked it - on describing it in one of my posts, I realized that any wealth coming into/out of a tier (and not just being redistributed within it) was being evenly distributed amongst everybody independant of income. I figured out, conceptually and sort of graphically, what needs to be done to fix this, but I can't seem to think of the math to do it with right now, and wanted to ask if you could help with that.
Basically, the new calculation does this... take average of 1/2 of every unit's Net (which I will now call InnerNet to be consistent), which may be a positive or negative number. (Do not yet take into account anything added or subtracted by the money the entire tier gets or owes from it's parent). Call this number the InnerBalance.
Now, the new part is, take the tier's OuterNet, the amount it gets or owes from it's parent, and divide that up across the units (say there are N units) such that if a unit's InnerNet is exactly average, they get or owe 1/N of the tier's OuterNet; and it scales linearly from there such that, if the tier's OuterNet is positive, the unit with the highest InnerNet gets 0/N of the tier's OuterNet and the unit with the lower InnerNet gets 2/N of the tier's OuterNet; or, if the tier's OuterNet is negative, the unit with the lowest InnerNet owes 0/N and the unit with the highest InnerNet owes 2/N of the tier's OuterNet. (This is the part I can't think of the math to do, although I'm pretty sure it'll involve a variable exponent in there somewhere to give a reciprocal equation depending on whether the tier's OuterNet is positive or negative). Call whatever number you get for each unit here, whatever they owe or are owed of the tier's OuterNet, their OuterBalance.
Add the InnerBalance and OuterBalance for each unit and multiply by percent of full time worked (as a decimal) and call this that unit's OuterNet. This is how much total that unit owes or is owed to it's tier. (I'll use the term Net now to refer to InnerNet plus OuterNet; basically one's net after taxes). Basically the problem before was that while there was a redistribution of wealth within the tier happening perfectly progressively, the actual tax or welfare part (depending on whether the tier has a positive or negative OuterNet) was being distributed flatly. So I made that part distribute progressively as well. Now the calculation should be an actual, exact median between a fully capitalist and a fully socialist system.
While I'm here, a few other thoughts I had:
"Full time" I had not previously defined. I think it should be dynamically defined as the average amount of time worked by units in that teir. That amount is also considered the "amount of time worked" by that tier itself. So say you have a lowest-level tier with 10 people, of whom 5 work 30 hours and another 5 work 50 hours (a week, of course). "Full time" would be considered 40 hours, and that tier would be considered by it's parent tier to "work" 40 hours (on average amongst it's population, you see). Then take the average of the tiers' "work" times for the tier's own parent to get it's "full time"... etc.
Also, if you have the time, do you think you could run the numbers taking into account this time-worked multiplier? When I was describing it the first time I realized that it might wind up with an unbalanced equation, more people oweing money than getting it or vice versa. If that was so, making full-time an average might fix it. Do you think you could check to see either way if it works out right?
And lastly, if you could, do you think you could help me phrase all of this in a nice, english legalese version of a mathematical equation? Like, just define what each of the terms are... InnerNet, InnerBalance, Net, OuterBalance, OuterNet.
One more thought that just occurred to me is that perhaps the Net of each unit from one period should be added to that unit's InnerNet for the next period. That way units who are still losing money total will get even more money the next period, because their negative Net from that previous period will be accounted for... and units who are still raking it in even after taxes will be taxed even more heavily. I worry that over time this will eventually socialize the entire system, though it would be an exponential decay in this "equalizing pressure" - the more equal people got, the less it would tax/welfare them and the more their own achievements would be reflected on them, so it might balance the system back out again. If it ever did become completely equal, which is shouldn't but we have to consider the possibility, everybody's Net for that period would be 0, and the distribution would spread itself right back out again the next period according to who's actually making the most money before taxes (since taxes are nonexistant if all Nets are 0).
Bah, too much socioeconomics for one night... off to find girls...
