Quantity theory of money: microfoundations

Question

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.

Answer 1

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

Answer 2

Well, I'm not sure the claim "raise M by factor k and P will also rise by factor k" is valid unless one also stipulates that V and Y are held fixed. Indeed, interest rate policy relies on the assumption that movements in key interest rates will shift Y more than they do P, with some of that action arising from a shift in V.

A two-good pure-exchange model (e.g., Edgeworth Box) is a pretty simple, micro-level model that would support the idea that when endowments increase uniformly, so do prices. Take one good as a numeraire, derive the prices in terms of the initial allocations and then increase the initial allocations by some arbitrary factor k. Relative prices won't change, suggesting that individual prices have risen by the same factor (be it equal to, less than, or greater than k). And to homo economicus, relative prices are what matters.

This relies on two aspects of the quantity theory: 1) the economy must be at equilibrium for MV=PY to hold, so that price ratios are already equal to ratios of marginal utility; and 2) increases to M by a factor k are manifested through an instantaneous increase to all agents' budget constraints by the factor k. As far as I know, this second point is only implied by the quantity theory's lack of prescription.