[A view of Dahka street crowds, Ahron de Leeuw]In order to attempt a solution to the problem, I chose to invent a unit that I call the MIU (minimum income unit), which I define as the amount of income an individual needs to stay alive. That would include nothing more than the cost of food and the least amount of medical care, shelter, etc. that a person can get by with.
The answer I've come up with, in case you can't stand the suspense, is this - Up to a point, the more money you have the fewer kids you need, but once people reach middle class, the more money they have the more kids they want.
Here is how I propose MIUs affect your ability and desire to have kids
1 MIU - If you earn 1 MIU, then you can't support anyone else. People might still form couples (i.e. get married), but they couldn't raise children or support their parents in their old age. Anyplace where the average income is that low would eventually fade away as people starved or moved away.
1 to 1.5 MIU - This means you earn a little more than you need to survive. If you have a partner, you fall just short of being able to afford to raise a child or support an elderly parent, and you certainly can't do both.
1.5 to 2 - Now a couple has enough money to raise a child until it's old enough to contribute to the household income, and at the same time, you can contribute a little to support elderly parents. Presumably , a married couple has no more than one dependent child at a time, and up to 4 elderly adults to support. If you earn, say, 1.6 MIU, for a total household income of 3.2 MIU, that means that you spend 3 on your immediate family with 0.2 left over for your parents. But your parents need 4 MIU, so you had better have 19 siblings and in-laws to help or your parents will starve. But people can only have so many kids in a lifetime. If we assume a reasonable upper limit is 10 per family, then the lowest sustainable income is about 1.6 MIU on average. Any less an the society will crumble.
3 to 4 MIU - At this income level you can raise a child until it's old enough to work while supporting 2 of the child's grandparents. You still need lots of siblings to support the other two grandparents. Each working-age child you have contributes to the household income, so you want lots of kids, but only one very young one at time. That could mean as many as 5 or 6 kids in a lifetime.
5 to 10 MIU - With this much income, you could pay someone else to take care of your child, you can support your parents, and you can put some money away for a rainy day. Presumably, you send your kids to school, and don't expect them to earn any money at all. That means that children are becoming an expense rather than income generators. If you have too many, they will eat away at your nest egg and you will be poorer in your old age.
10++ MIU - People who earn much more than they need to survive can afford to have lots of kids who live relatively luxurious lifestyles. The richer you get, the more kids you have, and some people will choose to indulge themselves by having lots, which makes the average number of children increase as people get increasingly wealthy.
To sum it all up, very poor people need many children to survive. As incomes increase, they need fewer, and have, fewer children. Once you're income is so high that you can pay people to care for and educate your children, they become expenses rather than sources of income, so you have very few, but they live good lives. Ultra-rich people can have more still kids because they can afford (and want) luxuries like mansions, expensive cars, and kids.
Although I haven't used much math, I am roughly estimating that the poorest people earn about 1.6 MIU each on average, will have as many children as physically possible, which is likely no more than 10 per couple.
People earning about 10 times the income they need to survive would have the fewest kids. People who earn either more or less than that will have more children. As it turns out, this sort of behavior has recently been documented, according to a 2009 article in the Economist newspaper.
While I don't want to delve into how the Human Development Index works, this graph shows roughly the sort of behavior I've described. The u-shape indicates that increasing prosperity leads to fewer kids, up to a critical point where more income leads to more kids.
To put more quantitative numbers on it, a graph from Gapminder World shows income verses fertility around the world as of 2009.
The people who chose to have the most kids earn about $500-$1000 per person, while people who chose to have the least number of children earned on average $10,000-$40,000. If you assume that MIU is about $500-$1000 per person per year, then those with incomes 10 or more times the MIU do indeed choose to have the fewest number of kids.
This is all very squishy math, and not really a very good choice for a Fermi problem, but it gets my point across - the best way to keep the population in check is to help lift the poorest people in the world out of poverty. Then you won't have to ask them to have fewer kids, they'll choose to have fewer all on their own.