I am expert on Mathematical economics, so for growing our QA site, we need to improve it by high qualified questions. So can I ask any high quality question that I know the answer?
Answering your own question is an acceptable practice in .SE sites. And indeed, it sometimes happens, when the OP eventually realizes the answer through the help he received, or also, deliberately, when the issue is considered "canonical" for the general subject of the Q&A site, and it is good to have it as a subject and also, provide a canonical answer. I have seen such threads, e.g. on the Cross Validated site. I would add also "corner subjects" that the "average" graduate student may not ever encounter -but PhD candidates and researchers will almost certainly stumble upon.
For example, taking the "Exercises" part of some advanced textbook, and start providing answers to them here, is not the way to go. But posting a question about even a part of a canonical model or solution method that it generally causes problems, lack of understanding, or opens new roads, is useful.
Out of my head, a question inquiring about some strange aspects that sometimes the Transversality condition exhibits in Optimal Control models with infinite planning horizon, and an answer showing how the strange result can come about (so, mathematically rigorous) and what can we say about it to make it more understandable, could be a planned Q&A that adds value to the site, because the Transversality condition is crucial to the solution and the internal consistency of the model.
Or a "question" about "what are the Fritz John conditions, and should we prefer them over Karush-Kuhn-Tucker in optimization problems with inequality constraints?" would offer the chance of an answer showing how the Fritz John conditions incorporate the (in)famous "constraint qualification" into the core conditions, and so provide solution points that the KKT conditions would be unable to detect (due to the failure of the constraint qualification).
So in general, yes, posting a question and then posting the answer in order to create "timeless content", is acceptable in principle and can also be useful, but the choice of such "questions" should be wise.