Would the health care production function eventually bend downward?
The production function for health care, like that of any other product, will eventually be downward sloping. This is because health care will experience diminishing marginal products as more inputs are added.
Think of this in terms of one given hospital. Our production function for that hospital will assume that the physical plant of the hospital remains fixed. This means that the hospital will have a constant number of rooms, MRI machines and other such things. Now look at what happens as we add labor to the hospital. As we start, the marginal product goes up. Hiring nurses helps the doctors see more patients at a given time. Hiring clerical workers allows the doctors and nurses to work more efficiently.
But after a time, new hiring no longer helps. If we hire more doctors than there are examining rooms, each new doctor will bring very little in the way of marginal product. If we hire more MRI technicians than there are machines, each new hire will bring very little (if any) marginal product.
This same dynamic will hold true for all hospitals and medical clinics in the United States in aggregate. Therefore, the production function for health care will eventually slope downward, just as it would for any other product.