MV ≡ PQ
Where M is money, V is velocity, P is price level, Q is economic activity.
Assume V’ is constant; then:
MV’ = PQ
Take logs (where lowercase letters denote log values):
m + v’ = p + q
Take difference with respect to a fixed date, and call that Δ :
Δm + Δv’ = Δp + Δq
Since V’ is constant, we can rearrange to obtain:
Δp = Δm – Δq
Let’s plot the change in the price level and the change in M2 to real GDP, relative to 2020Q1, the onset of the pandemic in the US.
Figure 1: PCE deflator (blue), M2/real GDP (tan), both in logs relative to 2020Q1 values. M2 values are end-of-quarter. NBER defined peak-to-trough recession dates shaded gray. Source: BEA, Federal Reserve Board via FRED, NBER, and author’s calculations.
It’s been about 2 years since the Fed increased high powered money and allowed that to manifest in an increase in broad money. Yet, the price level has only risen by about 7.9%, while the ratio of broad money to real GDP has risen 27% (in log terms).
This lack of correlation between broad money and the price level is not an artefact of the covid era. Here is a scatterplot of inflation against growth of M2/GDP.
Figure 2: Q4/Q4 PCE inflation vs M2/real GDP growth rate (calculated in log terms), and nearest neighbor fitted line (tan). M2 values are end-of-quarter. Source: BEA, Federal Reserve Board via FRED and author’s calculations.
A more sophisticated version of the quantity theory would allow trends in velocity, and/or interest sensitivity. I do not think this would be sufficient to make the quantity theory empirically more valid than alternative approaches (see comparison empirically ad hoc quantity theory vs. Phillips curve).