It should be noted that Vimes was specifically thinking of real generational wealth in that area. He'd spent some time in the home of a Lady from oldest and wealthiest family in the city, and saw that everything was old there, solid, built to last forever. Generations of clothing tailored to the fit of the family members, and saved if it was still in good condition, and if it was not, the fabric would be reused to make something else. Even the garden tools were old. The family owned multiple homes in various cities and in the country, and they owned sizable portions of the real estate in the city.
When you're part of a family like that, one who can simply give a building in a prime location away, without a thought (as she did, when new police station was needed, completely unasked), the only things you'd ever need to replenish would be perishable things like food and fuel. So you simply don't have to spend money on "stuff".
But the same holds true, though to a lesser and lesser extent the further down the ladder you go. Even a kid from a working class family who gets his first apartment with a parent cosigning, and grandma's old furniture and dishes and pots and pans is a big step ahead of another kid who can't get approved for that first apartment and had to pick housing that charged less up front but cost more over time.Personally, I'd revise Vimes' level 3 a bit: "The reason the wealthy stay wealthy." Wealthy families tend to stay wealthy. Rich people tend to blow their money rather than plan to provide for their grandchildren's grandchildren.