From $70 to over $20,000...wow. 

Wow indeed:

The grand total for his mother's six-day stay at Grant, for the use of the operating room, for his days in the nursery, for the various medicines and lab work -- for all aspects of his birth -- was $70 in 1947.

That's $726 in 2012 dollars.

So why does a healthy birth in 2012 cost almost $20,000?

1. Technology is expensive.

2. Malpractice insurance is expensive.

So what do we get for all that extra money spent?

Much higher survival rates: "In 1950 nearly 30 infants out of 1,000 died at or soon after birth; in 2009, that number was 6.4 out of 1,000."

Note at 6.x the USA is the worst in the first world.

http://www.oecd-ilibrary.org/social-issues-migration-health/infant-mortality_20758480-table9

I could not quickly find a similar cost comparison, but even ignoring the differing spread of costs borne by patient, government and insurance I'd expect the US to cost twice as much as the second most expensive.