The paper actually compares two particular models of fertility and says that one of them fits the data better. There's a very important variable that's omitted in both models - time and monetary costs of raising children, which are very hard to measure consistently across time and across countries. Without such consistent measures, no academic would write a paper, even though anecdotal evidence indicates that making raising kids easier - in monetary, time, and society acceptance's terms - might be very important. (That's what the BBC story linked in comments seems to imply).

If we were able to quantify those effects, it could happen that they would explain a bulk of the observed variability. Who knows - multiple regressions are tricky. A gang of aspiring Ph.D.s would tear apart the regressions in hope of proving the masters wrong, and at some point, we would have a pretty good idea of what matters and what doesn't. This is how science work - among other ways, by extending a hypothesis which is proved wrong at a later time. Or supported.

By the way, Boldrin, one of the authors of the paper, is well aware of the fertility dynamics in France and its possible explanations. I'm looking forward to an update of his paper.