Given the dearth of answers, I tried to do some more research. I found the following suggestive statistic:
In 2002, there was a baseline of 16,488 annual admissions to LCME medical schools; by 2009, the number of medical students enrolled had increased by 11.6 percent to 18,393... From 2002 to 2009 there was a 62.2 percent increase in annual enrollment [of osteopathic schools] - CGME 20th report
Given that allopathic and osteopathic degrees are legally equivalent, we might expect demand to be equivalent. So it is suggestive that AMA-accredited schools have much smaller growth in enrollment than non-AMA-accredited schools.
In fact, from 2002-2013, there will be 3k more DOs (an increase of 99%) and 3.5k more MDs (an increase of 30%). So again, unless there is just a huge shift in preference for DO vs. MD, it seems unlikely that this is the result of a free market.
EDIT: As far as I can tell, DO and MD are quite similar:
Osteopathic physicians, known as DOs, are licensed to practice medicine and surgery in all 50 states and have full scope of medical practice in over 50 countries... “We now find ourselves living at a time when osteopathic and allopathic graduates are both sought after by many of the same residency programs; are in most instances both licensed by the same licensing boards; are both privileged by many of the same hospitals; and are found in appreciable numbers on the faculties of each other's medical schools." - wikipedia
DOs are allowed to prescribe meds, do surgeries etc. the same as MDs. Further certifications (e.g. anesthesiology) are open to them just as if they were MDs. I don't think that DO is exactly equivalent to MD, but they seem similar enough that we would expect demand to be approximately the same.
The LCME accredits allopathic schools; it's made up of the AMA and the AAMC.
EDIT 2: this paper is more recent (from 2003) and claims that the AMA engages in rent-seeking behavior. It also gives as one of the reasons for the shortage a minimum wage requirement for residents:
The ACGME historically has required that teaching hospitals pay residents a reasonable wage and pay residents in all specialties the same amount. However, the wage that clears the market for residents in pediatrics or family practice may be too high to clear the market for surgical residents... I estimate that medical students would be willing to pay teaching hospitals for residency training in dermatology, general surgery, orthopedic surgery, and radiology. [As opposed to the hospitals paying residents]