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If you have 100 people and they all have some growth, so they all start with the same resources and it grows a little, then no matter what kind of distribution the growth rates have, you'll always end up with extreme inequality (the tail-end distribution we see today) after just a little time.

Math:

Reality

The only way in which you have zero or non-growing inequality is when all have the same growth rate which is just never the case. Even fluctuations in growth rates ultimately matter little, as long as you have some kind of growth rate distribution, you end up in an unequal state. This is also why you see tail-end distributions everywhere, in the size of countries, the size of companies, the wealth of individuals, and even the number of organisms from different species.

Not only are real-world growth rates distributed, they are also dependent on your current wealth. It’s e to the power of e to x. This is all capitalist critique summaried in one picture:

(I guess the more correct thing is something like this, with f(t) being time dependent wealth and g(f(t)) being a wealth dependent growth rate)

This translates to: very wealthy people have a higher growth rate than non wealthy people. The reason for this is also intuitive: If you're poor, you are struggling with survival. Everyone has a fixed burn rate they must spend on food, etc. If your return on growth falls below that your growth will essentially be zero or even negative. Yey.

But then there are billionaires who have effective growth rates of multiple percentages per year. I don't know what the calculations would be for Elon Musk, but it's probably pretty, pretty high. And now that he has made billions, his growth rate seems to be stalling a little bit, but it's still significantly higher than for most of the population.

One way to gain higher growth rates is, that talent - or more precisely the likelihood that someone will generate novel explanations - is also tail-end distributed amongst people. Rich people can spend more resources on finding and hiring this talent, which means that innovation becomes tail-end owned and innovation drives growth.

You could also just convert your wealth into political power and military force and just crush everyone elses growth rates.

In such a world, at the end of the universe, you have only one person who has everything, essentially the entity that kept the highest growth rate going for the longest will own 99.999% of everything. As we haven’t seen much decrease in inequality over history, I think this might actually happen.

Which is really bad. (Maybe??)

If everyone had the same amount of growth rate, but a different fixed burn rate, inequality also arises. So what if we put everyones burn rate to 0? We could also finance it by capping everyones growth rate at a maximum. This would allow everyone to catch up to the maximum growth rate eventually. You would end up with linear inequality at the end of times.

There is one effect which is quite interesting: monopolies or the largest structures are usually the least resilient. If you look at the senescence of trees (the likelihood of dying in a given year) it's basically constant, so they'll actually just never really die or age.

Unless they become the tallest one in the forest. Their death rate suddenly significantly increases to the point where you actually just die. This is mostly due to wind sheer, lightning strikes and and physical things. I don't know how this is for trees, but at least in humans there is also revolution, anti-trust laws or the David-vs-Goliath-effect (we favor small brands over large ones even if their products are worse). We collectively favor diversity over large growth rates for monopolies, so we turn their growth rates into S-curves. This is good, but I don’t think it is stable, fair, or enough. Everyone is probably worse off at the end of the universe, and inequality will not have gone away as much as people hoped.

Maybe we don’t actually need to restrain large entities, maybe bureaucracy, rot, and finite resources turn anything into S-curves:

I don’t trust this, someone will probably find a solution to all three, and see Googles stock growth rate actually didn’t really decline

Also, if we let this happen, they can still stifle everyone elses growth rate, like Russia just did.

This would also probably kill us all, because there is a dynamic where the one with the highest growth rate was probably the least careful when it came to technological innovation.

So no.

Maybe this is not that bad? If the single entity at the end wants to maximize well-being, is critical and non-authoritarian (never stifles someone elses growth rate, just improves his faster then anyone will ever catch up to) then maybe this isn’t that bad, right? Maybe diversity actually maximizes suffering, through allowing suffering to exist in the first place? Like, would you rather have 99.999% of the universe inhabited by happy people, or 50/50?

I have no clue how to answer this at the moment. The claim that diversity leads to more well-being is one of these claims that very few have probably thought through. A coral reef is most definitely a worse place for its inhabitants than a bee hive. Maybe. Ask me in a few years.

If everyone believes the universe will be run by one, it better be me, the one and only. I don’t know how, but we could probably all decide not to do that? This would essentially mean that we could have an equilibrium or an attractor state of a common growth rate (similar to the UBI). It shouldn’t be zero or negative for humanitarian reasons, but there is probably a game theoretic solution out there that works?

Yes, there is! At least maybe:

What you ideally want is situation where everyone realizes that when they optimizes their growth rate so that it maximizes the lifespan of the whole system, their total wealth is maximized. In plain words: making it to the end of the universe is hard, and even if you owned all of it, not worth it if it comes early.

So we need everyone to become growth rate optimizers, not maximizers. But any rational being that does not know how long it will last, should be one. Here is why:

Let’s take the 100 people from the beginning again. Same start, different growth rates. But they all have undefined lifespans and could live for millions of years.

Now what everyone wants is, that no one starts killing the others. Anyone might do so, to take everything, but if they start, others might copy him and then they might die. Bad solution. Everyone also wants technological progress to be as careful as possible, so no one releases a virus or similar. But they still want growth. So the optimal strategy is to find the optimal growth rate and then convince all the other ones, that their maximum is a tiny bit lower than yours. If the others are also rational they will all converge on one growth rate.

Which is quite interesting thing to think about. So yeah, the solution to inequality is immortality.