Sorry, but the whole idea is laughable. Not to mention the fact that it's another incentive to have more children and bigger families because the state will give you more money.
Should be easy to back that wild claim up with statistics. Can you?
Everywhere I look, I see the opposite. Families with a comfortable amount of disposable income have the 2.4 kids, but those in poverty (and particularly the working poor) have 3, or upwards. Anecdotal, but I'm yet to see evidence to the contrary
If you try and slice it by family you find that the 'traditional' family makes up a fairly small percentage of the population.
Richer men in the same country have more kids.
PS: I think I just misread your comment. By "national scale" I thought you meant in the same country.
I don't know of any study on national level, i.e. of families in a single country. The only thing I know of are anecdotes and stories about very poor families having 5+ children.
EX: http://econlog.econlib.org/archives/2011/06/kids_are_normal....
Higher income > more kids, Higher education > fewer children.
This seems fairly obvious as people are less likely to have children and stay in school. But, as I understand it the effect is larger than that.
http://economix.blogs.nytimes.com/2011/07/01/richer-people-w...
http://freakonomics.com/2011/06/10/the-rich-vs-poor-debate-a...
Basically if you have two groups of American born, collage educated men, with of the same age then income positively correlates with number of children. But, if you compare collage dropouts vs. people that have PHD's then the PHD population makes more money and has fewer kids. So, education is a huge confounding factor.
We went from 14 being a reasonable age to start a family to 24 or even 34 being the 'reasonable age' that's a huge impact. But, ‘wealth’ as an independent factor aka adjusted for age, education, country etc becomes a positive factor.
PS: Historically, starvation also limited family size.
I always suspected that the reality was more like "people who are able to afford children and able to make long-term plans tend to have only as many children as they want, when they feel they are ready, but people who aren't able to afford children or who don't make long-term plans will tend to take a more devil-may-care attitude towards procreation."
Since population explosion has felt out of fashion, there's a long time that I didn't see such plot. But what I've seen has a much clearer tendency than most social correlations people use at real decisions. It's about as clear as most correlations of quality of life with GDP.
A talk by Hans Rosling[1] on the subject.
But bear in mind: Statistics could indicate the extent that the incentive is having on the incentivized population group, if at all.