In several recent cases, for example in New York and Colorado, courts have stepped in to mediate cases where a patient received a big balance bill from an out-of-network provider. They ordered hospitals to accept amounts far closer to what they agree to from in-network private insurers or Medicare.

“This is the amount they are legally entitled to collect,” said Mr. Richman.

Dr. Lopez’s bill came after he sought help at the emergency room following excruciating abdominal pain. Sent home with pain medication, he awoke hours later to a phone call from the hospital: Come back! A review of his tests showed he needed surgery. He didn’t stop to ask if the hospital was in his network, or for a cost estimate.

So, in an example like that, is there mutual assent?

Hospitals say yes, that signed admission forms, which include a promise to pay, constitute mutual assent, even if there was no price disclosed.

No, counters Mr. Richman. If a tax preparer provided no upfront estimate, he could not suddenly bill a client for $10,000 if the going rate for the service was $1,000 or less. The higher fee would never hold up in a court of law, since there was no “mutual assent” about price.

But what, if anything, should Dr. Lopez offer to pay? What is reasonable or average in a system where the price of a hip replacement can range from $15,000 to $150,000, or a blood test can be $5 to $500?

Based on the hospital’s list prices, Dr. Lopez’s bill came to nearly $21,000. His insurer, Cigna, using a formula it said is similar to what Medicare uses, said the maximum it would cover was $11,160. It paid 80 percent of that lower amount, and Dr. Lopez paid the remainder. Baptist Hospital is billing Dr. Lopez for nearly $8,000 more, saying it wants the full charges.

“I’m an economist,” said Dr. Lopez, who teaches at the University of Memphis. “I understand how abusive these practices are. There is not a single market price.”