What is the basic difference between being a researcher in a corporation, and being a researcher in a university?
Is there really any difference apart from giving lectures and grading student's exam papers?
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As a researcher in a company, I need to take issue with the current answers as reflecting a very narrow view of corporate research. This view is common in academia and reflects the typically limited experience of university researchers with the diversity of the non-academic research ecosystem.
A large amount of research in corporations is, indeed, focused on the near-term needs of products. Companies whose business model is based on widget sales will tend to put their money into better widgets, after all. There are a lot of business models, however, that are not about widget sales. Some examples:
The life that you will lead as a researcher in these different sorts of organizations is very heterogeneous. Even within a single organization, such as the one that I work at, the goals and incentives are very different for researchers on the primary investigator track, the program manager track, and the implementer track. Even tenure can be less different than you might think, given the existence of fellows programs and the like.
So what is the basic difference?
The goals of research are different; a researcher in a corporation is expected to find solutions to specific problems that impact the functioning of the corporation. Alternatively, the corporate researcher may work towards inventions that add direct value to the corporation. Often this means developing something that other corporations do not have.
On the other hand, a university researcher is more free to choose problems that appeal to curiosity, provided the solutions add to the intellectual stature of the university. The focus would be on adding to the reputation, not profitability of the university (though reputation would indeed draw more grant money and generate profits, and therefore baby, bathwater etc.)
A more concrete difference is that universities generally value output in the form of research communications (journals, conference proceedings etc). Corporations generally place higher value ok intellectual property markers like patents/design patents.
It is sometimes argued that university research is disruptive due to its unfettered nature, while corporate research is incremental due to being confined to more norms and standards. This is true in some fields and less in others.
The basic difference is that research in corporations is much more likely to be very applied, even just applied to product development. In universities (and government labs) the research is more likely to be basic research: answering questions for the sake of knowledge itself.
This isn't entirely true and some (a few) corporations do basic research but it has declined quite a lot in recent decades.
There are a couple of reasons for the split and the change, but one is that, in recent years, there has been an extreme focus in corporations on returning value to stockholders and company officers are rewarded for that and little else. This was not so prevalent in the past.
Another reason for the split, is that most government funding of research (again, not all) goes to universities and government labs, so that the stakeholders are the general citizenry, not stockholders of a corporation. So, the questions supported by government funding are more likely to be basic research questions: "what are quasars doing anyway".
Some of the exceptions might be pharmaceutical research that depends on basic knowledge before you can start to think about products. Agricultural research may be similar, but it isn't my field.
Bell labs was once the paragon of company based basic research, as was IBM research. They are mere shadows of their past glory.
Some companies are supporting things like research into encryption and AI at the moment, and some of that is fairly pure math, but they are doing so primarily for competitive reasons, not for the love of just plain "knowing stuff". Tenured faculty have few pressures to produce for competitive reasons, other than those related to reputation.
Another difference, that will matter to some, is that the pay (base salary) in corporate research is likely to be higher than in a university for people with similar qualifications. One friend of mine earned just about double what I did and it would be hard to find many distinctions in job expectations between us. He was mostly paid to think and his ideas were often useful. He was, in part, a walking advertisement for the quality of work at his companies. I think my retirement plan was much (much) better than his, but his salary was much higher. We were both quite satisfied with our decisions and could arguably have swapped positions.
Speaking from personal experience. I've done both kinds.
As a professor, I looked for questions that intrigued me and that I thought I could answer. Sometimes they came from conversations with colleagues, from papers I read, from courses I taught, even from the time I spent playing mathematics with elementary school kids. When I found new mathematics that was publishable, I wrote papers. I changed fields often, as different things caught my fancy. Once I had tenure I felt no pressure to publish. There were fallow years when I focused on teaching and wrote textbooks (and on some administration).
As a consultant at a software company that built products that depended on the mathematics of queuing theory I learned a lot of queuing theory. Sometimes the answers I found to problems ended up in the products. Sometimes they led to talks or papers at professional conferences in the industry. Very occasionally they led to a papers in math journals. That was a bonus. If my work hadn't been useful either in products or as advertising they would not have kept me on.
I've worked as both a researcher in academia and industry. Most answers are correct in that it varies widely by position. However, here is my experience.
You pick the question, though there are constraints based on available funding, advisor, department goals, etc. It's freedom in the sense a tagged wild animal is free; people watch you closely and want interesting things to happen but the animal can go where it wants. You'll also need to determine if the question is noteworthy or novel in some way. Once you get a result, you also need to communicate/sell the result in academic writing, which has a very established formula with limited freedom to deviate. Other forms of selling your research help, such as conferences, presentations, etc. My view here is "pre-tenure". Salary is generally lower than industry.
Your boss picks the question, though a great boss will be open to hearing better questions and ideas. It's not as important to make sure the question is novel/noteworthy, a well run company has already made this determination. Once you get a result, you need to convince people it matters. There is much more freedom here than academia - just set a meeting and "sell". The downside is that audiences tend to vary widely in technical acumen where in academia most people are experts in the field. Salary is generally higher than academia with potentially lucrative bonus potential.
I moved from industry to academia, and I'm still in the process of deciding where I'd like to end up. I like teaching, which is a nice perk of academia. Industry requires many other 'non-research' tasks, too. Some are fun like mentoring and some are a waste of time like 'town halls'. There are pros and cons to both. If you're a brilliant researcher, it won't matter where you are (Einstein was working in a patent office when he published some of his most groundbreaking papers, after all).