MW ≡ PQ
Where M This is money, AT is the speed P price level, Question is an economic activity.
Suppose V’ is constant; then:
MV’ = PQ
Let’s take the logs (where the lowercase letters denote the values of the logs):
m + v’ = p + q
Take the difference with respect to a fixed date and call it Δ:
Δm + Δin ‘ = ∆P + Δd
FROM AT ‘ is constant, we can rearrange to get:
ΔP = ∆m – Δd
Let’s plot the change in the price level and the change in M2 to real GDP relative to the first quarter of 2020, the beginning of the pandemic in the United States.
Figure 1: Deflator PCE (blue), M2/ real GDP (tangent), both in logarithmic units relative to the values of the first quarter of 2020 M2 values at the end of the quarter. Peak-to-trough dates as determined by the NBER are in grey. Source: BEA, Federal Reserve Board via FRED, NBER and author’s calculations.
It’s been about 2 years since the Fed increased the high power money supply and allowed it to show up as an increase in broad money. However, the price level rose by only about 7.9%, and the ratio of broad money to real GDP increased by 27% (in log terms).
This lack of correlation between broad money supply and price levels is not an artifact of the coronavirus era. Here is a graph of the spread of inflation in relation to M2/GDP growth.
Figure 2: PCE Q4/Q4 inflation vs. M2/ real GDP growth rate (calculated in logarithmic terms) and fitted nearest neighbor line (tangent). M2 values at the end of the quarter. Source: BEA, Federal Reserve Board via FRED and author’s calculations.
A more sophisticated version of the quantity theory would allow for trends in velocity and/or sensitivity to interest rates. I do not think that this would be sufficient to make the quantitative theory more empirically valid than alternative approaches (see below). comparison of empirically ad hoc quantity theory with the Phillips curve).