I commented on my own post about the fact that MB/NGDP ought to be the primary variable to describe an economy based just on dimensional analysis and quickly realized that the equation of exchange is just
MB/NGDP = k P MB/NGDP = k, so, well, duh. [Thanks Mike for catching the typo in the equation.] But that sent me down the rabbit hole of trying to show a graph that captures the picture in my head. The best result was this graph of the price level versus the monetary base:
The graph has the (normalized) data for several countries (US, EU, Sweden, Australia, Japan, Canada as well as the US 1929-1944 as colored points) along with the fits (dashed lines) to the function:
We can see that as the monetary base grows, the price level flattens out. Note that the model fit lines actually happen on a 3D surface (you can see they sidle back and forth a bit in places), so here is the same data along with the 3D plot of the function above (σ = MB/MB0):
Since κ is in a very tight range (roughly κ = 0.6 to 1.0) you end up with what looks like a line in the first graph at the top when you graph P versus MB. Interestingly, you can rotate the 3D image to make all the data points fall on a better line (I removed the surface for clarity):
Is this showing a universal behavior of economies?