Comparing the fixed quota system with a tradable procreation permit, I would argue that the latter is more questionable than the former. Even though a tradable procreation permit would be efficient whereas a fixed quota system would not, its impact in terms of eroding people’s perception of how fair their community is offsets efficiency gains. People’s expectations about one another affect their behavior towards the group, a position underpinned by a large body of recent academic evidence that rejects the idea of the utility maximizer representative agent. It is rather the case that emotion and moral codes play a major role in human decisions. In this vein, one’s attitude toward his or her group, including whether one cooperates or not and to what degree, depends fundamentally on the person's expectations about the rest of the group. Hence, the fairness of public policy fosters cooperation among agents. The design of public policy and institutions must account for this behavioral effect, taking the discussion beyond efficiency concerns.
Specifically as regards the options at hand, a fixed quota system is problematic in a highly unequal world. Imposing a significant fine on the poor for an extra child limits their freedom regarding family decisions, whereas for the rich the cost of having more than one child dissipates in their mammoth budget. A fixed quota system, however, can be socially acceptable by letting the fine on extra child be proportional to income. On the other hand, when there is a market for the right to have a child, such as a tradable procreation permit, the nature of having children changes drastically. When kids are goods, having a child implies an opportunity cost absent under a fixed quota system. Kids would most likely become luxury goods under such conditions; rich families would be notable for having five kids. For the average guy, having a child would imply not earning a lot of money, as the right to have kids would presumably be quite expensive. A young couple would thus be faced with the awkward decision of having a kid or selling the right to do so and travelling around the world, buying a car or an apartment and so forth. Who knows what the price of life would be? For poor people, having a child would be a burden; selling their fundamental right to have a family would be a matter of making ends meet. A market for children introduces an opportunity cost, its corollary is to create a fallacious equivalence between money and life, as if they were interchangeable.
Most economists would argue it’s a paradox that one system can be more efficient yet less preferable than other. Mainstream economics isolates moral concerns from its analyses claiming everything can be understood as goods. Economic models and its restrictive hypothesis, however useful, must operate within its boundaries. Most relevant economic questions lie beyond orthodox framework, and, in such cases, dialoguing with other fields of science is paramount.
PS: I'd like to thank the old salt Pedro Mendes Loureiro for very helpful comments on a draft of this post, thank ye matey!