A few of my friends believe that the cure for cancer already exists, but the pharmaceutical companies won't release it because they make more money treating the disease rather than curing it. In my head, the company that had the cure would end up making the most because they would have a better product. My questions is which model would be the most profitable?
You have put your finger on an important topic that economists have been arguing about since Kenneth Arrow’s analysis of the incentives to innovate in “Economic Welfare and the Allocation of Resources for Inventions” in _The Rate and Direction of Inventive Activity_, edited by Richard Nelson (1962). As the subsequent discussion illustrates, one can, with different plausible assumptions produce situations in which the incentive will be either to introduce the superior innovation (cure for cancer) or to hold it off the market.
If there were just one pharmaceutical firm and it controlled all of the treatments and acted like a monopolist then it might indeed be reasonable to imagine the scenario you have described, holding back on a cure for cancer because the profits of selling existing drugs would be greater. However, the reality is more complicated, since there are multiple competing firms, each of which controls only a small portion of the market. So the gains from curing cancer would presumably come both from displacing the firm’s own products but those of
Its competitors. This should tip the balance toward introducing the drug. Further, since in the scenario you describe the inventor cannot have patented their discovery, they would have to worry that it would leak out or another company would arrive at the same solution, and then they would get no benefit at all.
Indeed, we see something like this in more mundane markets, where pharma companies compete to introduce marginally better drugs than those already on the market, because they can profit by capturing a large share of the market from other companies.
Given these considerations I think it is unlikely that anyone could develop a cure for cancer and not introduce it into the market.