In my experience, outward appearances aren't a great indication of whether someone is truly wealthy. Sure, there are people who like to flaunt their money, and certain purchases can't realistically be made without having some money (e.g., if your home is worth a lot of money, then you're probably not poor, though some people own large homes that they've only paid off a small fraction of), but many of the people I can think of with established wealth don't tend to flash their money too much. I would almost say that the people I've known who tend to flash money the most are the ones with high incomes, but very low net worth and practically no savings. Many of these individuals do quite well when times are good, but the moment they lose their jobs, their kids get pulled out of private schools, they immediately give up their country club memberships, and sometimes, they have to sell off their house and downsize. Personally, I don't find high expenditures without savings to represent true wealth. If you can't afford to continue your existing lifestyle (and especially, fund your kids' education--that's something I personally think is important) for at least a while if you lose your job, then you're not rich. And for the truly rich, I tend to envision people with enough wealth, and enough income from sources like dividends, etc., that they can sustain a relatively high-end lifestyle indefinitely without needing to actively work (though they may, of course, enjoy working and choose to pursue a stimulating career).
As for monetary cutoffs for being "rich," I'm not sure. $20+ million, maybe? $10 million is getting pretty far up there too. To truly get to the point where you don't have to worry about expenses at all (including the price of property you buy), I generally think of $100+ million. (And I've heard of properties that cost nearly $200 million, so maybe that metric isn't even good enough, but I'm pretty sure that I could do whatever I wanted to financially with $100+ million.)
I don't think of myself as rich, and I have a feeling that I'll never amass as much money as I've suggested is necessary to be truly rich. ($100+ million certainly isn't looking likely, especially since I'm leaning towards career choices that probably will cut my income in favor of greater intellectual stimulation. And at this rate, I'm not going to make a decent income for over a decade. But hey, at least I'm not in debt at the moment, so I consider that to be a good start.)
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