If I am not mistaken, people in the past tended to have enough offspring so that they increased the population of their family until the maximum sustainable population was reached, due to the lack of birth control and what have you. Since the upper classes tended to have large amounts resources, why didn't they keep increasing their population until they reached the maximum sustainable population, which would make them quite poor? Did the upper class simply not stay in power for long enough for them to become overpopulated? Did people not want to reproduce as much as I think they did? I haven't found anything discussing this after googling it.
The reason that the upper class was able to preserve itself was because only a handful of the descendants obtained most of the property of the founders.
For instance, Genghis Khan was prolific in his production of children, to the point where perhaps 0.5% of the world's people are descended from him from him, or at least have his DNA. In a world of 7 billion people, that would be 35 million people.
On the other hand, only four descendants through his wife Bortei inherited the Mongolian Empire after he died. (It was split into four pieces.) And those pieces were later (mostly) recaptured by China, Iran, Russia, and Kazakhstan. Unless you were a descendant who "made a deal" with one of those four countries (or Mongolia) to keep an even smaller chunk.
The upper classes who stayed wealthy did so because of their economic practices. The global economy today is a recent invention; economies tended to have less interaction on a broader scale in the past. Thus, there was an even more immediate feel of the zero-sum game (my gain comes from your loss) in ancient times.
One way for a person to make the initial climb above his neighbors was to charge interest or, if he was in the right position, tax. One charging interest on a loan makes money simply because he had some money to begin with, not because he is actively doing anything with it. Experts agree that interest rates in ancient times were quite steep.
If someone in the village had a good harvest and his neighbor had a poor one, the one who was better off could loan his neighbor some of the produce and charge interest when it was paid back. This was one of the ways people got the upper hand at the beginning. Considering that most people grew little surplus, this meant, eventually, that the lender all but owned the land his debtor worked for him.
This obviously balloons. The lender is able to lend out more money and collect more interest. With careful exploitation, he eventually has everyone in the village indebted to him and receives a portion of everyone's harvest without having to do any work himself.
The upper classes didn't always stay wealthy. Those who stayed wealthy were the ones where the heirs paid attention to how their ancestors acquired and maintained their wealth.
I diverge a little, but it was the intent of the Mosaic law to prevent this exploitation. Debts were forgiven every seven years, slaves were freed every seven years, and lands were returned every fifty years. It was an economic reset to prevent class disparity.
"In the past" is a little vague. I'll concentrate on the West because I have no acquaintance with this in the New World and little with Asia
In 500 BC, when there was no primogeniture in Athens, men controlled their wealth through infanticide. Since everything had to be split between sons & a dowry provided for daughters, families consisted of two sons and one daughter, or less.
On the other hand, in England in the 1890's, the tradition of "the heir and a spare" was well-established (To Marry an English Lord by Wallace & MacColl) and a man's idea of "birth control" might consist of keeping a mistress.
One stays wealthy by hanging on to resources, not by divvying them up into parcels too small to maintain one's class. For a peasant, lots of children are wealth, because they are free labor for many years. The wealthy/noble seemed to have figured out fairly early that their family would stay wealthy only if they kept it fairly small. After all, once primogeniture was established, younger sons often had no way to maintain themselves. They might serve a lord (like their brother) or go into the Church, but were regarded almost as a nuisance.
The first class to start using birth control regularly was the upper classes, because of higher education & sophistication, & having the money to keep buying condoms (in their primitive form, dating back to the 1600's).
Roman upper class families, especially senators, restricted their family size in order to avoid having their sons fall out of senatorial status. This risked having all the children die, which led to the practice of adopting children from other families to inherit the name, and property, of a noble family that has no heirs.