The veterinary evidence strongly suggests that the rate of growth of health care costs in the U.S. is not explained by our health insurance system. We generally for pet's health care completely out of pocket, yet from 1984 to 2006,

health care spending per person rises by 350 percent, vet spending per dog rises by 335 percent, and vet spending per cat rises by 340 percent.

So, if you want to cut costs, go after the absolute level of spending, not the rate of growth. (Unsurprisingly, Robin Hanson has radical proposals for how to achieve this.)