Nornagest comments on Economics/demographics question: If a child unexpectedly dies, how much does this shrink the next generation? - Less Wrong Discussion
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"Constrained" may have been a poor choice of words, but I still don't think that's a good assumption. It's plausible that people are more inclined to produce children when they expect them to be economic assets, but outside of a fully exploited subsistence environment I don't expect that to depend linearly on population. And even if it did turn out to be linear, I don't think our instincts are accurate enough to make that determination without something extremely simple and obvious (like remaining land area for farming) to cue off of.
Note that richer countries have fewer children per capita than poor ones, and richer demographics within those countries tend to fewer children than poorer ones. There's some confounders like infant mortality to deal with, but even so this would be strongly antipredicted by your theory.
(On the other hand, I'd be willing to believe that a given set of parents is likely to want a certain number of children, and to produce more up to that number, fertility allowing, if one unexpectedly dies. Inheritance and other family-controlled resources can play into this, but it remains a purely local phenomenon.)
I can think of several reasons offhand why rich people would have fewer children:
The first factor applies to the degree that becoming rich is controlled by the parents' actions. The second applies to the degree that being rich is associated with the parent's traits. And the third applies to the degree that the parents want to achieve a particular level of wealth for their children and need to spend money to do so. Having the children become marginally richer because of factors that are not related to the parents' actions or traits and do not involve spending money on them would not lead to that marginal increase being correlated with having fewer children.