Employers may write the checks to the insurance companies, but workers still pay for the coverage they get from those employers. The total cost of an employee is what matters to businesses, and fringe benefits are as much a part of compensation as cash wages. When health costs rise, firms don't become less competitive, as if insurance were lopped out of profits. Instead, nonhealth compensation drops. Or wages rise more slowly than they otherwise would.
A recent study from none other than the White House Council of Economic Advisers notes exactly this point: If medical spending continues to accelerate, it expects take-home pay to stagnate.
It's certainly true that the