This is getting interesting (the monetary base from inflation and aggregate demand)

I took the system of equations
$$ \frac{Q^{d}}{Q^{d}_{ref}} = \left(\frac{Q^{s}}{Q^{s}_{ref}}\right)^{1/\kappa} $$
$$ P = \frac{1}{\kappa}\left( \frac{Q^s}{Q^{s}_{\text{ref}}}\right)^{1/\kappa - 1} $$

And numerically solved for $\kappa$ and $Q^{s}$ (the monetary base) using the empirical nominal GDP for $Q^{d}$ and  the CPI (less food and energy) for $P$. Here are the results: 

 

The first graph shows the model (blue) for $Q^s$ and the St Louis Adjusted Monetary Base from the FRED database (gray); the second graph shows the model for $\kappa$ (blue) and the empirical estimate
$$
\kappa \sim \frac{\log MB}{\log NGDP}
$$
(the latter in gray).

I did the same thing using the GDP deflator instead of the CPI and got a similar result

Nothing Earth-shattering here. They give remarkably similar results and the results largely match the the empirical data. The interesting aspect is that both formulations miss the rounds of quantitative easing occurring since 2008. The index $\kappa$ has a more well behaved monotonic increase (my opinion -- $\kappa$ seems like it should be slowly varying).

However, it is pretty cool that now we have only NGDP and the price level as inputs to the model instead of the previous results which used the empirical $\kappa$ so is based on the monetary base, the NGDP and the price level.