What do universities get from students with valuable research ideas? I recently stumbled upon an obscure economic theory. Currently, I'm trying to understand whether or not it makes sense. If it does, I intend to build a certain simulation of the economy (using some concepts of that theory). This software will allow to better understand the economy than existing approaches. Let's assume that everything went fine and I do have a simulator of the economy in my hands, plus (someone else's) theory which explains, why it works.

The next step would be to find a scientific institution (university), which could check my findings and officially confirm that they are scientific (look through the stuff I wrote, find logical errors, verify that this is not some hocus-pocus pseudoscience etc.). Thereafter it may be much easier to spread this knowledge. A little book published by a nobody (even if it contains breakthrough knowledge) won't be noticed and its wisdom won't be applied. When the same information has been verified by some prestigious university, it's a different game (there is more trust in these findings).

If I want the university to invest resources in such check (and do it better than many professors who don't even read the theses of the students they supervise), they must have some incentive. Let's rule out money (paid by me) as such incentive.

What do universities get from sound theories developed by people who study or cooperate with them?

In other words: Imagine, some of that university's professors looks through my writing, checks how the simulator works and comes to the conclusion that it, indeed, does what I promised. No fatal flaws are discovered in the work and they agree that this method is valuable (should be researched further).

How exactly does the university, its professors, and other people doing research and teaching there benefit from this discovery?

My answers:


Better rankings. If this discovery is truly valuable, it may increase the university's prestige and help them get more (or better) students, money from public and private sponsors etc.
The professor may publish a paper (together with me). If the findings are truly new, the paper may get cited and improve the professor's h index.
The university may get consulting contracts for applying my findings to client's problems.


Notes:


I need the answer to this question in order to determine whether or not I have a chance of "selling" the cooperation with me to a university. You can only sell something, if the buyer's (university's) benefits are greater than the costs.
I have very little academic background therefore the answers above may be wrong.
Your question is based on a few assumptions, some of which are loose or close to be false. I'd outline some of them:


Ideas are valuable to Universities.  No, they are not. Universities may be interested in hiring people who have ideas and are able to convert them into working implementations and/or academic publications, thus generating prestige for the University and attracting more grant money and more students. But if the University does not have people to get your idea onboard, your idea, however cool, is wasted on them.
Universities can check my findings and officially confirm that they are scientific. No, Universities don't do this. The only `official' routes to recognise the scientific contribution is to publish it in peer-reviewed journal or patent it. Both require much more effort on your side than just formulating an idea.
Professors look through my writing, checks how the simulator works and comes to the conclusion that it, indeed, does what I promised. Normally, professors would not spend time doing this unless they see a clear benefit in terms of publications and grants. This basically means that someone checking your work would expect to become a co-author of this work and share all the prestige (and potentially profit) generated. Think twice if that's what you want.