And so American patients are stuck with bills and treatment dilemmas that seem increasingly Kafkaesque. The hopeful news is that American health care spending has grown at a slower pace over the past four years. While that is partly because of the recession, economists say, many credit the cost-containing forces unleashed by Obamacare with a significant assist. Even at that rate, many models suggest that nearly 25 percent of gross domestic product will be eaten up by health care in 20 years. That is not sustainable.

“It’s like a diet you can’t just stop, because it’s starting to work,” said Michael Chernew, an economist at Harvard Medical School. “And remember, we haven’t even lost weight yet, we’re just gaining weight more slowly.”

Many health economists say we must move away from the so-called fee-for-service model, where doctors and hospitals bill every event, every pill, every procedure, even hourly rental of the operating room. Though insurers try to hold down costs by negotiating discounts or limiting reimbursement, this strategy has limited power because armies of consultants now advise hospitals on what is known as “strategic billing”: Losing money from trauma patients? Hospitals can add on a $10,000-plus “trauma activation fee.” Medicare not paying enough for a broken wrist? Add a separate “casting fee” to the bill.

“People in fee-for-service are very clever — they stay one step ahead of the formulas to maximize revenue,” said Dr. Steven Schroeder, a professor at the medical school of the University of California, San Francisco.

Given that national or even regional rate-setting is out of the question, most health economists argue that the nation needs a new type of payment model, one where doctors and hospitals earn more by keeping patients healthy with preventive care rather than by prescribing expensive tests.

Such models exist: A number of hospital and doctors groups engage in so-called capitated care, where they are paid an annual fee by an employer or individual for all patient needs and must work within that budget. The Affordable Care Act promotes a strategy focused on accountable care organizations, in which similar networks can earn financial rewards for figuring out how to save money while meeting standards for good care. But such models are still far from the norm in a country where a majority of physicians are in business for themselves and doctors and hospitals bill separately.