What are the main questions in monetary economics? This post isn't really about the answers to those questions. It's about the questions themselves.
This is how I think about dividing up monetary economics into a small number of main questions, and how those questions are related to each other. Others may want to divide it up differently. Sometimes the questions you want to ask may depend on what sort of answers you have. And how those questions are related to each other may depend on how you answer them too.
This is partly a response to Robert P. Murphy's EconLib article. I think about it a bit differently. In particular, I see the equation of exchange and the cash balances approach as just different ways of formulating the same question.
Here are my three (or maybe four) questions:
1. Why do people use monetary exchange? Why don't they use barter, or a central clearing house where all goods can be exchanged between all people at once (a Walrasian auction), or some other method of exchanging goods? Why do we (mostly) converge on a single good to use as a medium of exchange? Closely related questions would be: if there are two different media of exchange, what determines where one will be used rather than the other; how does one medium of exchange get replaced by another?
2. How much money will individuals choose to hold (what determines the demand for money)? How quickly will individuals choose to turnover money (what determines the desired velocity of circulation)?
Those sound very different, but I see them as being two ways of asking what is fundamentally the same question. Both are asking about the relation between stocks of money and flows of monetary expenditures. Both are asking: why don't people spend new receipts of money instantly so that velocity is infinite and the stock held is zero? Why is the desired inventory of money strictly positive and desired turnover of inventory of money finite?
This second question assumes the first question has already been answered. If people didn't use money, they presumably wouldn't want to hold a stock of money.
3. What determines the value of money? (What determines the price level?) What determines the rate of inflation (the rate of change of the price level)?
If you are a quantity theorist (understood in the broadest sense), your answer to this third question is closely related to your answer to the second question. Your answer to the second question gives you a theory of the demand for money. You add in a theory of the supply of money (which will depend a lot on what particular good is used as money and on the particular monetary institutions of time and place). Putting demand and supply together you get a theory of the value of money (the reciprocal of the price level). But if you are not a quantity theorists you may see the second and third questions as unrelated.
4. (I can't decide if this is really a separate question or just a part of the third question). What ensures that the equilibrium value of money is positive? If a good has a value of zero, people can't use it as money (question one), and so there will be no demand to hold a positive stock of it (question two), and so it won't have a positive value (question three).
If the good had some other use, apart from being used as money, it might still have a positive value regardless (provided the non-monetary demand exceeded supply at a price of zero). And yet most of us nowadays use an intrinsically useless good (like paper) as money. Does paper money have two different equilibrium values, and if so, what stops us observing the equilibrium in which the value is zero?
Those aren't the only questions you could ask. What are the effects of monetary policy? What is the best monetary policy? How valid are economic theories which abstract from monetary exchange (which is most of economics)? Etc.
Worthwhile Canadian Initiative 
The unit of account is the key attribute of money because there are numerous historical examples of unit of account only money. Either the money is notional like Carolingian shillings and pounds, or it is symbolic or temporary like the stone money of Yap. The larger pieces of stone money never circulate, since they are much to heavy to move, in fact, that is part of what makes some stones valuable.
The store of value and medium of exchange aspects are, past a certain point, in conflict, because of the opportunity costs of holding one form against another.
he exact nature of that co-determination depends, inter alia, on the monetary policy regime.
It depends on the distribution of wealth, monetary policy regime, fiscal policy, desire of debtor households to increase borrowing, whether firms are willing to understake risky investment, why people save, expectations of future profits, the birthrate, and what the Chinese are doing 😛
And in some cases, the monetary policy regime is either impotent, or less effective than the others.
You have to be careful appealing to supply and demand.
1) There’s no necessity for the supply and demand curves to be independent or stable.
2) There’s often a possibility of short circuiting, that is manipulating the curves. Business schools teach courses in this.
3) Supply and demand can fail in the face of perverse incentives where cost has been shifted (like medical costs).
4) Supply – a stock can behave differently from supply – a willingness to supply, which is a function.
5) The shape of a demand curve doesn’t have to be very well behaved (e.g. monotonic), nor need it be solely a function of price (even allowing for supplies with different utilities).
and this list isn’t exhaustive.
Peter N.: “The larger pieces of stone money never circulate, since they are much to heavy to move,..”
Are you sure that’s right? Ownership of the stones can circulate, even if the stones themselves don’t. We don’t always have to physically put money in our pockets. Simply saying “I will pay you my stone, the third one on the right is mine, for your cow.” should do it. Much like a cheque.
Houses don’t move either, but they still “change hands”. IIRC, the German Imperial authorities once confiscated all the Yap stones, by painting a black cross on them, to signify ownership.
Victor: there are lots of different stores of value. Money is a small part of our total wealth.
The Murphy article is a beautiful distillation of much of what’s wrong with modern economics as taught.
“The original and more intuitive version of the equation is MV = PT, where M stands for the total quantity of money in the economy, V is the “velocity of circulation” (meaning how many times, on average, a dollar bill changes hands for the time period in question), P is the average price of a transaction (“price level”), and T is the total number of transactions.”
“The problem with stating the equation of exchange in this form is that the term T includes all transactions, including the sale of previously-produced goods and even of assets… However, economists typically don’t welcome this result because they want to exclude financial transactions from the analysis in order to focus on production.”
“Consequently, it is more common nowadays to see the equation expressed in the form MV = PQ, where Q stands for newly-produced goods and services (“real output”) during the period in question, and where both V and P are restricted to transactions involving newly-produced goods and services. In this updated form, the equation of exchange can be very useful to illustrate certain relationships.”
So M = PQ / V and M = PT / V and PT / V = PQ / V. But what does this mean? Nothing. The assumption that you can separate the “productive” economy from the total economy and deal with it in isolation is nonsense. It’s no more reasonable than trying to define the economy of all left handed people and thinking that the result was of any use.
“For example, the equation shows that if the quantity of money in circulation doubles while velocity and real output remain constant, then the price level doubles, too.”
It does nothing of the kind. Given the implicit assumption here that M and V in the two equations are the same, we get P T = P Q, in which case we can’t be “restricted to transactions involving newly-produced goods and services”. So M and V must be different, but then why does an accounting identity in M and Vp imply an actual economic relationship with another identity in M and Vt. It doesn’t.
Unfortunately, there’s more to come.
“In the cash-balance approach, the economist recognizes that in equilibrium—though it may be a fleeting condition—it has to be true that every individual is satisfied with the amount of money he holds.”
“Having thought through the necessary and sufficient conditions for a typical individual to be in equilibrium with respect to his cash holdings, the economist finishes the analysis by saying that the sum of all desired cash balances must equal the total quantity of money. In equilibrium, it must be the case that the community collectively desires to hold the exact amount of money in existence at that moment—no more, no less.”
This is underwear gnome logic. The economist begins the analysis and ends the analysis. What about the middle part where all these fleeting equilibria occur at the magic moment when we aggregate?
Equilibrium is one of the most abused concepts in economics. Realistic models never produce a single stable equilibrium. No one ever has or will ever prove that the real economy is in equilibrium, because, of course, it isn’t.
Given this result, the author shows how various economic theories might be demonstrated. However, since it is well known that given a single false statement in logic, it is possible to prove anything whatsoever, it’s not a very impressive showing.
This is undergraduate toy economics, a genre that has outlived its usefulness (if it ever had one) and has become outright dangerous.
In the words of Einstein “”Everything should be kept as simple as possible, but no simpler.”
The stone money of Yap seems to be a form of special debt money, of a sort discussed by Graeber. It doesn’t circulate, it acts as a debt marker. The debt is repaid, and the money ends up back with the original owner. Graeber has all sorts of examples of unusual forms of money.
Some core questions are missing.
1. How does the changing flow or demand for money effect production & consumption choices?
2. Why do different polities have different monies, and what establishes the relations between these different monies and what consequences do these changing relations have on production and consumption choices.
3. How do laws governing various money regimes come into being, and who do these laws benefit, and who do they harm?
These seem to go beyond your 1-4, but are of central interest when one turns ones attention to question raising patterns involving money.
A classic sort of pattern giving rise to scientific inquiry and causal explanation is the attempt to account for the origin of a thing, eg the source of the origin of species or the origin of biological adaptations.
This question is a bit different, and closer to a genuine empirical scientific problem than is your own #1.
Often it is helpful to go back to the original patterns that raised questions & to the theoretical constructs that helps us see those patterns, or that made patterns problematic, to uncover and highlight and flesh out the deeper, empirical scientific problems to be explain, ie the problems that a capable of genuine explanations in terms of causal mechanisms.
As much as anythings, Darwin’s great achievement was in perceiving and recasting the problems to be addressed in biology — problems which came to light in tandem with the causal mechanisms that could account for the patterns imagined without reference to magic or divine intervention.