Despite competition and choice in the private insurance system, Medicare spending has grown more slowly than private insurance premiums for comparable coverage for more than 30 years.Below is a chart showing the annual change in cost. At first glance it supports the contention that private insurance on average has higher growth rates. The actual annualized rates from 1969 - 2009 are 8.2% for Medicare and 9.5% for private insurance. That makes a big difference over 40 years.
But check out the spike in the late 80's. Up to that point spending growth tracked fairly closely, the annualized growth rates from 1969 - 1986 are within a tenth of a percentage point. Then from 1987 to 1991 private insurance becomes a house on fire. What happened? The answer is that Congress raided Medicare using the newly enacted PPS reform. From the Incidental Economist piece:
Aggregate Medicare hospital payment-to-cost ratios fell every year from 1987-1992 because hospital did not restrain costs as quickly as payments were adjusted (Guterman, Ashby, and Greene 1996). As Medicare margins fell, private pay margins grew over this period. The effects of managed care had not yet been fully felt in the commercial market, leaving private purchasers vulnerable to hospitals’ market power...Hospitals took back from the private insurer market what they lost from inadequate Medicare payments. Hospitals pushed that as long as they could until employers responded by embracing managed care, initiating the age of HMO's. Getting back to the growth rates, if private insurance had grown at the same rate as Medicare from 87 to 91 the annualized rate over 40 years drops to 8.7%. Still half a point higher then Medicare, but a lot closer then Medicare/single payer advocates like to claim. Especially if you think cost shifting happened in more then just those 5 years.
More important though is that there is no fiat answer to slowing cost inflation. From 1969 to 1986 Medicare and private insurers were equally bad at it, competition or its absence was irrelevant. Once payers started getting tough both systems proved capable of containing costs. PPS made a huge difference to Medicare, and managed care proved effective in private markets (look at 92 - 97).
What it comes down to is that for cost control you don't need any particular system, you just need the backbone to stand up to providers.