The national government recognizes the problem these communities and really the whole nation faces and have come up with a bunch of action plans to try to turn things around. Local governments sometimes kick in additional amounts. This usually more than covers the cost of having a baby in a hospital. In fact I have heard of people actually making money on this by going to a mid-wife who charges less, shopping around for hospitals, reducing their stays in the hospital so reducing costs, having their husbands bring them boxed meals during their hospital stay, and many more ingenious cost reducing ways to have a baby and make some money. This might even be a good business for any experienced midwives out there.
Some localities give more, some give less.
Even if the child is born with a serious birth defect and needs constant hospitalization. They are discussing raising this to the first five years or until the child enters elementary school, but this may be a few years away. Then on top of this the government will give you 5000 yen per month for 5 years for having the child. If you have two, you get an additional 5000 yen per month. For your third child they will give you 10,000 yen per month and this amount for each additional child.