I'm pretty sure this is John Brunner's "The Vitanuls".
It has an semi-immortality pill (not perfect and just rolling out):
"I think I follow you," the matron said at length. "I take it the anti-death pill is a success?" ... "You in the rich countries like America and Russia have been trying for years to find a broad-spectrum specific against aging, and I think--knowing your countries by hearsay--that there must have been a loud angry argument over who should benefit first."
It takes place in India, which gives them the setup to bring in reincarnation as a Chekov's Gun:
The matron gave him a chill stare. "I think that is uncivil of you. The teaching of Hinduism is that the soul is born again, throughout an eternal cycle, until it achieves one-ness with the All."
I can't provide direct page quotes for the ending (I was quoting from Google Books and they omit the next few pages), but a summary does indeed indicate that babies do start getting born with a curious lifelessness, and it is the lack of enough human souls that is the cause.
And there is this line:
"No, that's absurd. You can't mean to maintain that we've run short of human souls, as though they were stored in a cosmic warehouse and issued off the shelf every time a child is born!"