When Mr. Reid presents his shoulder to his own orthopedist in Colorado, the doctor is quick to recommend a shoulder replacement. It will cost his insurer tens of thousands of dollars (assuming it agrees to pay), with unknown co-payments for him.

In Japan, the foremost orthopedist in the country (waiting time for an appointment, less than a day) offers a range of possible treatments, from steroid injections to surgery, all covered by insurance. (“Think about it, and call me.”)

In an Ayurvedic hospital in India, a regimen of meditation, rice, lentils and massage paid for entirely out of pocket, $42.85 per night, led to “obvious improvement in my frozen joint,” Mr. Reid writes, adding, “To this day, I don’t know why it happened.”