The wealthy don't breed like jack rabbits because they don't need to.
Some people, mainly religious people, will have stupidly large families for their own reasons.
Most everyone else though in the middle class keep it to a few children though.
In primitive and agrarian societies, there is a direct pay off for having children. They are labor. How costly that labor ends up being depends on a lot of factors, but they are a source of labor that someone (their parents) own and can control. They need this extra labor so they can provide for themselves better, not have to work as hard and have someone to take care of them as they age.
Rich people don't need any of that. Unless someone is breeding for the purposes of a family dynasty, they don't need the labor. They just hire people. They don't need help in their labors; they work as hard as they want to, not need to. They don't need someone to take care of them.
Unless someone's objective is to out-breed [insert political/religious/ethnic group here]s, having "as many children as you possibly can" simply isn't worth the pay off for all the pain and, even though they're rich, sundry time and costs to be a decent parent. (Doctor's visits, preschool, summer camp, etc.....x12.) The pay off of warmth and a connection to your kin even diminishes the more kids you have. Listen to any stand up comic talk about coming from a large (6+) family and what you generally won't here is anecdotes about a high degree of parenting.