The setup you describe could either be something very open and clearly visible, but assuming your greedy domineering people were also fairly intelligent, they might as well follow a somewhat more covert approach.
It is not necessary for the slave to know he is a slave. In fact, he might be a lot easier to handle (and to be made do the things your ruling class wants him to) when he thinks he is in fact a free man, and maybe even had a chance to become a member of this fabled and famous rich ruling class.
To make sure he stays where he is, though, he heends to be led in many small ways.
First, make sure his struggle is just hard enough for a big majority never to actually improve their situation. Make sure housing is expensive, so are schools.
Also, make sure that your common slave is constantly told that he needs to buy all the nice stuff your companies are making, in order to be recognized as one who might actually climb the social ladder.
Most of them will fall for that, and of course they cannot really afford all those thinigs. But since they see them as an investment into a brighter future higher up in the social ranks, it is perfectly acceptable to just borrow the money. (From your banks. Or those of your fellow ruling class members).
Of course this lifestyle is exhausting. But hey, the slave knows that it is only until he becomes a member of the top tier, so that is okay, also, holidays in just the right places (so his fellows know that he is on the way up the ladder) are almost entirely inside his budget, and the commercials keep telling him how perfectly relaxing they are!
The right gadgets for the children, of course, are absolutely necessary. First, to compensate for the time your slave spent trying to climb ladders instead of playing with the kids, and also, of course, to signal to his fellows how close one already is.
Noo, no worries. Those mortgages are no problem, and the credit card balance is perfectly under control! As soon as the next rung of the ladder is climbed, all this will be paid for in no time!
In the meantime, the ruling class can play golf, take their yachts for a cruise, and laugh their backs off watching the futile struggles of the slave race.
Of course, this is entirely fictional. Such a setup could never be established among modern, civilized humanity.