Money has two defining functions: it is the medium of account (all prices are quoted in terms of money); it is the medium of exchange (all other goods are only bought or sold for money). ("Store of value" is not a defining function of money, because my canoe is a store of value too.)
Scott Sumner argues that it is the medium of account function that matters. My view is different. Here is my view:
Demand and supply of the medium of account determine the equilibrium price level.
Demand and supply of the medium of exchange determine whether the economy is in a boom or a recession.
[Update: Scott responds.]
Here is a simple model, where one good (gold) serves as medium of account, and another good (silver) serves as medium of exchange.
[Update: just to clarify terminology: in my model, gold is the medium of account; and (say) an ounce of gold is the unit of account.]
The third good is haircuts. Haircuts come in different styles. Styles of haircuts are located on a circle. Each person can only produce one style of haircut, located at one exogenously fixed point on the circle. Each person prefers some other style of haircut, located at some other point on the circle. He gets less utility from a haircut the further away is the style from his most preferred point on the circle.
This silly story does three things: it motivates trade in haircuts (you don't like your own style of cutting hair); it motivates monetary exchange in haircuts (you probably don't like the style of someone who likes your style, so barter won't work unless everybody gets involved); it motivates imperfect competition (so you can get booms as well as recessions, since price will be above marginal cost in equilibrium).
There are two markets: a market (more exactly, a continuum of markets for different styles) in which haircuts are traded for silver; a market in which gold is traded for silver. Nobody trades haircuts for gold, because:
Gold is worn as jewelry. It's a Veblen-good, where what counts is not the physical quantity of gold worn, but the value of that gold in terms of haircuts. "Look at me! I'm wearing 10 haircuts' worth of gold around my neck!" There is a demand for a real stock of gold, which depends positively on income from selling haircuts.
Silver has no other use except as a medium of exchange. Because people meet at random, they need to keep an inventory of silver in case they suddenly meet someone who can cut their preferred style but don't meet anyone who prefers the style they can produce. There is a demand for a real stock of silver, which depends positively on income from selling haircuts.
All prices are quoted in ounces of gold. Just because that's the custom (and I can't think up a silly story to motivate it.)
The exogenously fixed physical quantities of gold and silver are G and S, the price of haircuts in terms of gold is Ph, the price of silver in terms of gold is Ps, and annual income from haircuts is Y haircuts.
Full equilibrium is defined by three equations:
1. G = k.Ph.Y (supply of gold = demand for gold)
2. S = k.(Ph/Ps).Y (supply of silver = demand for silver)
3. Y = Y*. A third equation which determines the equilibrium number of haircuts produced, depending on the marginal utility of getting a haircut, the marginal disutility of giving a haircut, and the markup of price over marginal cost, which depends on elasticity of demand for a particular style of haircut.
Note that I have assumed that the demand function for gold is identical to the demand function for silver. The only difference is that Pg=1 by definition.
Note also that I have used the Cambridge "k", rather than the Fisherian "V", approach. Rewriting the equation as S.V=(Ph/Ps).Y makes sense for silver, which circulates as the medium of exchange. It does not make sense for gold.
It is obvious that equations 1 and 3 alone determine the equilibrium price of haircuts Ph. The only role of equation 2 is to help determine the equilibrium price of silver Ps.
But what determines Y when we are out of equilibrium, because the Emperor Diocletian has issued an edict forbidding any price changes?
Thought-experiment 1. Start in equilibrium, hold all prices (update: both Ph and Ps) fixed, then halve the stock of gold (medium of account). What happens?
There is an excess demand for gold in the gold market, but nothing else happens. The market for haircuts continues as before. People want to sell some of their silver for gold, but they can't, because nobody wants to take the other side of the trade. The production and sale of haircuts for silver continues just as before, because there is no change in the relative demands for silver and haircuts. There is no change in the marginal utility of silver, the marginal utility of getting a haircut, or the relative price of haircuts and silver Ph/Ps. So trade of silver for haircuts continues just as before.
An excess demand for the medium of account does not cause a recession.
Thought-experiment 2. Start in equilibrium, hold all prices (update: both Ph and Ps) fixed, then halve the stock of silver (medium of exchange). What happens?
There is an excess demand for silver. People want to sell some of their gold for silver, but they can't, because nobody wants to take the other side of the trade. People want to sell more haircuts for silver, but can't, because nobody wants to take the other side of the trade. People want to buy fewer haircuts for silver, and they can, because nobody can stop you buying less of something.The quantity of haircuts bought falls, until the marginal utility of haircuts increases by the same proportion that the marginal utility of silver increased when the stock of silver fell. The quantity of haircuts sold will halve.
An excess demand for the medium of exchange causes a recession.
The medium of exchange is different from other goods, because everybody both buys it and sells it. If you want more, and can't buy more, because everybody else wants to do the same thing, you just sell less. And when people sell less medium of exchange, they buy less other goods, which means a recession.