I live in Ontario, Canada where we have a single payer health care system. But this system, as good as it is, does not cover everything. So many working Canadians also get supplemental health care insurance through their employer (it is a sought after benefit, one that unfortunately I do not have). This supplemental insurance covers things like dental, physiotherapy, massage therapy, or a private hospital room.
My thesis is that for these supplemental services insurance actually serves to inflate the costs of these services and allows providers to earn excessive salaries.
My main experience has been with dentistry. Currently the going rate for a check up and cleaning is roughly $150 per visit. So for a family of four going twice a year that would be about $1,200. Not an insignificant sum. The cost of something like a crown is also expensive - about $1,250.
Even though there seems to be an excess supply of dentists prices are kept high by rates negotiated between the dentist association and the major insurance companies. Essentially those without insurance are left paying higher rates because those with insurance keep rates high.
By way of comparison I was recently in Taiwan, which does cover dental under its health plan. Because we did not have coverage we had to pay the going rates. For 2 cleanings and a crown we paid a total of $350 (in Canada the same thing would have cost $1,550). Now if you are thinking I would never go to a dentist overseas let me tell you that the office and service were well above what we typically get in Canada. In addition I had some other medical work done there and it too was done more efficiently and with better equipment than at home - quite a surprise for me.
Here is what seems to happen. Because most individuals do not have to pay the full costs of their supplemental health care services there is much less effort to keep costs down. Insurance companies are less concerned with costs than you might think. They are happy as long as they can pass on the costs ... and companies really have very little negotiating power as the insurance companies offer similar rates because they all have roughly the same negotiated costs.
I do know that if people did not have coverage they definitely consume less - in my case I may go to a dentist only once a year or even far less often. Without insurance dentists would clearly have less work at current prices and may as a result have to cut prices to attract customers. It is impossible to see how prices could be higher without an insurance system in place.
To be clear, my argument is that private insurance increases prices, especially for non-essential services. I firmly believe that a single payer system remains the most efficient and equitable way to provide health care.