Besides providing for you in your old age, children have economic effects here and now. In a subsistence economy, a child is an asset very early in its life- even a four or five year old can look after chickens. Subsistence agriculture requires a lot of labor, and the cheapest, easiest way to get it is to grow it. There's a clear, short term economic incentive to have more children.

Compare to the developed economy. In our world, a child is a liability, not an asset. Children do not work, but they still eat. They demand video games and designer clothing. Then you send them to college. When it's all over, maybe they'll support you in your old age, but our customs aren't nearly as strong in that respect. Your kids might just put you in a nursing home and forget about you. This is why developed countries often experience ZPG.

The really key question about population growth, then: Will the birth rates fall fast enough to give us ZPG before population grows beyond sustainable levels? This in turn hinges on how quickly the Third World develops, which is a complicated question for another node. At any rate we seem to have some time, since we can easily grow enough food to support a considerable increase, if we could manage to distribute the food to those who need it.

As Azure Monk points out, every increase in longevity will destabilize this again. The question here is whether agricultural technology will keep pace with medical technology. If it doesn't, there will be problems. In the extreme, science fiction case, that of true immortality, we would of course require a zero birth rate in order to have ZPG.