[T]o bring up such children might be an unbearable burden on the family and on society as a whole, when the state economically provides for their care. On these grounds, the fact that a fetus has the potential to become a person who will have an (at least) acceptable life is no reason for prohibiting abortion. Therefore, we argue that, when circumstances occur after birth such that they would have justified abortion, what we call after-birth abortion should be permissible.The authors essentially claim that the birth event is not morally relevant in the decision to terminate a pregnancy. If abortion is morally permissible, it's a non-sequitur to hold that infanticide (the authors object to the use of this word for rhetorical purposes) is immoral, so long as the only salient difference in the decision is that the organism is now located outside of the womb.
Swiftian pro-life arguments are fun to contemplate, but they'd be a lot better if they refrained from making explicitly fallacious appeals to fiscal externalities. The "burden on society" argument is one of public choice. That is, the level of care of the poor or infirm funded through public expenditure is made by the sovereign. If welfare payments are the problem with "burdensome" children, advocate for reforming the welfare system. The median voter only supports poor families by express political choice.
Of course, a Straussian reading of this piece would suggest that this frippery is what the authors want you, dear reader, to cluck your tongues at, but this is in the Journal of Medical Ethics. My priors are that they're sincere on the fiscal externalities point. More's the pity.
But on the chief argument: does birth impart full moral status on a person? Or is there more of a spectrum starting at conception and increasing logarithmically to adulthood (however that might be measured)? If early infanticide were completely legal, what would be the actual outcome? Do you suppose there would be a rash of baby murders o'er the land?
Raising children is not euvoluntary (for infants—think of their BATNA). Let's not compound the issue by confusing legitimate externalities with politically chosen "externalities". It's so rare that it's practically unheard of for a healthy human to be a net lifetime negative externality on society, and when he is, it's probably because he's chosen either a life of crime or of politics.