Sometimes people (at least on twitter/X) talk about how AI companies being publicly traded earlier is really great for power/wealth inequality. This quantitatively obviously doesn't make sense and there is a simple argument for this. To grow wealth by X companies must gain that much in additional (pre-dilution) valuation relative to the point of investment. To reduce inequality you need many, many trillions and companies just aren't that valuable (yet). I do agree it's helpful for a financial reduction of concentration of power if companies are eventually publicly traded in a normal way, but this only makes a big difference once companies are a large fraction of global wealth (e.g., $150 trillion). Being publicly traded might also help with concentration of power for other reasons (e.g., improved transparency due to SEC requirements, better shareholder governance), but this is a separate argument.
Quantitatively, doubling the wealth of the poorest 50% requires ~$5 trillion. So valuations must be way over $10 trillion for this to even be possible. (The wealth of the bottom 50% in just the US is roughly the same as the wealth of the global bottom 50%, so the numbers are basically the same for the wealth creation needed to double wealth.)
The numbers are more extreme for the poorest to reach the wealth of the current richest: the wealth of the median person is ~2000x lower than the top 0.01% wealth. So reaching within 10x of this would require ~200x-ing the median wealth ($15,000 trillion in total). AI will grow wealth this much (at least PPP adjusted), but many more doublings are required.
Of course, AI companies being public doesn't result in the ownership distribution being that much more quantitatively egalitarian than it is today. And there's no reason that the stock market massively growing would directly reduce (rather than increase) wealth inequality. (Being publicly traded is presumably somewhat better for wealth inequality than being private if these companies are were most of the growth is coming from.)
Absurd amounts of growth in AI like reaching >>$150 trilllion assumes that AI is massively undervalued (at least: in expectation, conditional on humans retaining control, and conditional on private markets being allowed to own this) due to the markets not pricing in a singularity (which I think is true).
If you think markets are not pricing the singularity is true and >>$150T valuations are possible, why "This quantitatively obviously doesn't make sense"? (I read your sentence as "This is obviously not true", maybe you mean "This isn't obviously true"? Claude thinks the most natural interpretation is the former.)
The "permanent underclass" idea from the tweet seems to be mostly a concern of American white collar workers -- and so I imagine the claim here is more or less that this lets relatively wealthy American white collar workers who are vulnerable to being automated away buy some insurance against that. Post-IPO, a white collar worker with a couple of tens of thousands of dollars available can probably buy that insurance via leveraged positions in the stocks, thus escaping the "permanent underclass."