Quantity theory of money: microfoundations The quantity theory of money states the price level is proportional to the amount of money in circulation. That is, if the quantity of money increases by some factor $k$, the price level will increase by the same factor $k$. My question is simple: in which economic models should one expect such a claim to hold true?

To be clear, I am asking about 'microfounded' models which contain full descriptions of the preferences and constraints of the underlying agents (including but not restricted to general equilibrium models). I am not interested in equations like $MV = PY$ unless you can show that they arise from such a model.
In the chapter 4 of the graduate textbook  Lectures on Macroeconomics there is "The Overlapping Generation with Money". In that case we have a model which is microfounded (i.e. accounts for how our consumers interact with money on the microeconomic level). This is essentially an explanation of Paul Samuelson's 1958 "consumption loan model".

You can find a useful set of videos on this model here:
https://www.youtube.com/playlist?list=PL10zuK3j9SNl15Xy-EGteGDTd74j4uE_n