A delicate balance, part 1

Nick Rowe puts forward an interesting analogy using a car's speed and speedometer to make his point about equilibrium conditions and causality. If the car's speed is S and the location of the needle is N, then in equilibrium, aS = bN. In the analogy S is assigned to inflation and N is assigned to the interest rate. He goes on to say that increases/decreases in S cause increases/decreases in N, but not the other way around -- at least not in the intuitive way. Rowe points out: "if I grab the speedometer needle, and rotate it clockwise, this will not cause the speed to increase and the gas pedal to go down". In fact, he goes on to say "that when [the Bank of Canada] wants the car to increase speed it turns the speedometer needle counterclockwise, which is the opposite direction that the equilibrium relationship would suggest."

My immediate response was: what kind of equilibrium is this?

One type to check is thermodynamic equilibrium. Effectively, if macroscopic variables S and N are related by aS = bN in thermodynamic equilibrium, the set of different microstates with macrostate S must be equivalent to the set of different microstates with macrostate N. The different microstates include situations where everyone's financial situations are re-assigned to different people, for example (much like trading the positions and speeds among identical particles). However if we change N to N', changing the microstate, S would have to change to S': if the new microstate had been already in the equivalence class S, then N' would have to be equal to N. And vice versa. This is how, e.g. entropic forces work. To maintain equilibrium, a delicate balance of changes in microstates has to be occurring.

So the aS = bN equilibrium can't be a thermodynamic equilibrium if it doesn't work both ways.

Another type to check is mechanical equilibrium where the balance of forces on an "object" cancel, leaving no net force and therefore no acceleration. Non-conservative forces like friction exist and can produce outcomes like the kind Rowe mentions above. It could also be the case that aS = bN represents an unstable mechanical equilibrium, per Paul Krugman. I believe an unstable equilibrium is consistent with Rowe as well, but I am not sure because he doesn't say what happens if you turn the needle clockwise. In this picture, an increase in N leads to a decrease in S and vice versa. In this way, your object starts to move away from the unstable equilibrium where aS = bN. 

But!

Where does it move to?

In Rowe's picture, S gets bigger as N gets smaller, making aS' > aS = bN > bN'. Now the entire point of Rowe's argument is that the economy at (S', N') doesn't experience a force to return to (S, N) (which is what Steve Williamson is saying). That leaves only three possibilities: 

1. (S', N') represents a new stable equilibrium cS' = dN'
2. There is a force directing (S', N') to a new stable equilibrium (S'', N'') such that cS'' = dN''
3. The economy never reaches an equilibrium (wheeeeeeee!!!!)

Interestingly, none of these situations are aS = bN which Rowe (and Scott Sumner) say the economy should return to in the long run. Now their explanations give me reason to believe that they really saying the economy will actually go towards cS'' = dN''. Switching back to the underlying economics for a minute, there are two ways economic agents could potentially get rid of unwanted cash:

• Somehow the agents make holding cash look more attractive by lowering inflation (this is what Williamson is saying and is consistent with considering aS = bN a stable equilibrium). Krugman says he needs to see the "somehow" story to believe it.
• Agents buy goods and services with the cash which should cause inflation (this is what everyone else besides Williamson is saying and is consistent with considering aS = bN an unstable equilibrium)

The second choice, if it stops, represents the path to the new stable equilibrium cS'' = dN'' in 2) mentioned above; there will be a new inflation rate S'' and a new nominal interest rate N'' which can't be (S, N) because then S, N would have been stable we would have used the first choice to get rid of the cash.

One way out of this conundrum is that the economy is actually a different economy in the future (for one thing, it's larger) and the condition aS = bN at time t1 is equivalent in some way to cS'' = dN'' at time t2. That's entirely possible, but I prefer a different way out: the equilibrium aS = bN does not exist.

That is the subject of part 2.