This is (supposed to be) a simple "teaching" post. There should be nothing new here. Just a newish way of presenting old stuff. The more interest-elastic the demand for money, the more important is the distinction between temporary vs permanent changes in the money supply.
Take a standard money demand function, where the demand for money is proportional to Nominal GDP (unit elastic with respect to nominal income), and depends negatively on the nominal interest rate i, with an interest-elasticity e. And assume the nominal interest rate is a positive function of the expected growth rate of NGDP. Set money demand equal to money supply Ms.
Ms = NGDP.i-e where i = r* + E[(dNGDP/dt)/NGDP] . Something like that.
Suppose you increase the level of the money supply by 1% for T periods (people expect the increase to last for T periods), then reduce it again to its original level. NGDP jumps up, then slowly returns to its original level over the next T periods. How much will Nominal GDP increase? (What is the elasticity of the initial change in NGDP with respect to the change in Ms?)
That depends on two things:
1. How long is T?
At one extreme, if T is forever, so it's expected to be a permanent 1% increase in Ms, NGDP will increase by 1%. So the elasticity of NGDP with respect to Ms will be one.
At the other extreme, if T is (expected to last for) a fraction of a second, NGDP won't increase at all. So the elasticity of NGDP with respect to Ms will be zero.
The bigger is T (the more long-lasting the increase in Ms is expected to be), the bigger will be the elasticity of NGDP with repect to Ms.
2. How interest-elastic is the demand for money?
At one extreme, if the demand for money is perfectly interest-inelastic (e=0), a temporary increase in Ms will have the same effect on NGDP as a permanent increase in Ms.
At the other extreme, if the demand for money is perfectly interest-elastic (e is infinite), a temporary increase in Ms will have the same effect as an increase in Ms that lasts only for a second.
For any given finite and strictly positive T, the more interest-elastic the demand for money, the smaller the elasticity of NGDP with respect to a temporary change in Ms. (Draw a vertical line on my diagram.)
To say the same thing another way, the more interest-elastic the demand for money, the more permanent a change in Ms will need to be, to have the same effect on NGDP. (Draw a horizontal line on my diagram).
3. Putting 1 and 2 together, we should get a picture something like the one I have drawn above. Except there's a whole family of curves, one for each interest-elasticity, and I have only drawn two members of that family.
It's a bit more complicated than that, because the interest-elasticity of the demand for money will change over time. But I think the diagram still shows the important point: the more interest-elastic the demand for money, the more important is the distinction between temporary vs permanent changes in the money supply.