Jaime Rosenthal, a senior at Washington University in St. Louis, called more than 100 hospitals in every state last summer, seeking prices for a hip replacement for a 62-year-old grandmother who was uninsured but had the means to pay herself. The quotes she received might surprise even hardened health care economists: only about half of the hospitals could provide any sort of price estimate, despite repeated calls. Ms. Rosenthal's grandmother was fictitious, created for a summer research project on health care costs. But the findings, which form the basis of a paper released on Monday by JAMA Internal Medicine, are likely to fan the debate on the unsustainable growth of American health care costs and an opaque medical system in which prices are often hidden from consumers.