Medicine and pharmaceuticals would be better. The reason is the public sector has an incentive to create blockbuster drugs that manage to cure expensive or orphan illnesses (illnesses that affect a small number of people or that affect poor people overseas). Serious illnesses that can be cured/controlled via medication will save money in public health costs, and improving health overseas will allow foreign economies to grow and create new markets. Plus the public sector can gamble and lose hundreds of millions, while private enterprise cannot take those risks. Also the public sector can do more basic research which may not be profitable immediately but which may come in handy 20 years later.
Pharmaceutical companies devote much of R&D to creating knockoffs of pre-existing drugs to get around patents. According to Marcia Angell, most of the basic research that goes into true groundbreaking drugs is done by the public sector. The private sector mostly works on tweaking pre-existing drugs to remarket them in new formats.
In this alternate world there probably wouldn’t be a half dozen plus versions of statins, antipsychotics or SSRIs, which would have some drawbacks (the side effect profile varies for each drug so having a dozen different anti-psychotics or statins to pick from has benefits). However we’d probably have more treatments for TB, malaria, tropical diseases and possibly Alzheimers, schizophrenia, etc.
Using cash rewards (like the X-prize) is a good idea. I believe Hillary Clinton ran on creating a public reward system to fund innovation into solving world problems. That would be good because you get more R&D than the cash prize. I think the x-prize for space travel and 100mpg cars created $100 million in R&D to win a $10 million prize.
There was a book written by an MIT professor I glanced at once. He basically claimed it was best to separate pharma industries into 2 parts, the R&D and the marketing. Let the R&D be funded by public sector funds and prizes, and then let the marketing side market those drugs. He felt that was a better way to get blockbuster and meaningful pharmaceuticals to the public.
As it stands the ‘market’ demand it to tweak pre-existing drugs for diseases people in wealthy countries have and repackage them to create new patents. In his world there might be a $20 million prize to a pharma R&D company for coming up with a novel class of chemicals that can fight malaria, and those chemicals would be leased to a pharma marketing company.