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I've read through questions like: Implications of declining money velocity and noticed this trend has intensified since, with velocity printing the lowest readings in history in the past few months. Currently, velocity stands at 1.13.

https://fred.stlouisfed.org/series/M2V

From what I gather, the mainstream interpretation is that the low velocity levels are a result of monetary stimulus making its way into M2. The expanded balance sheet of the Fed was mainly aimed at stabilizing financial markets rather than the real economy. With interest rates near/at zero, cash replaces short-dated bonds as the preferred risk-free liquid asset. In this sense, velocity reflects growing demand to hold cash versus use it for transactions in the real economy.

I haven't found any literature on sub-unity velocity, so I'm left wondering if there is some sort of singularity that cannot be crossed without violating a sacred economic theorem. However, given that:

$$ V = GDP/M2 $$

It would seem that there is nothing in the laws of physics (indeed, nor economics) that would serve as a fool -proof guarantee that GDP must exceed money supply. Surely with Weimar Germany and other hyperinflationary episodes, the ratio slipped into sub-unity territory. I concede it may be an extreme scenario for much of the developed world, but in the face of supply chain disruption and quarantine measures, it merits a thought experiment.

Question

Hypothetically, if the US was to fall into sub-unity velocity, would there be a clear interpretation and how much significance would the act of crossing below one have? Optionally, what if V was negative?

(I realize not everything can be taken to the extreme in econ. As an example, take negative prices. Instead of customers paying firms for goods, they would pay firms not to produce. Or whatever the case, the interpretation of negative prices isn't clear.)

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Hypothetically, if the US was to fall into sub-unity velocity, would there be a clear interpretation

$V$ is by definition the average number of times 1 unit of currency is used (See Fed explainer on that). A clear interpretation of $V<1$ is that there are some units of currency that were never used during the measurement period. If an hypothetical economy consists of 2 1\$ bills, 1 will be buried in the garden and never used and the second one only used once to buy goods and services worth of 1\$ the velocity of money would 0.5 (you could either calculate this directly as $V=\frac{1+0}{2}$, or indirectly in this scenario $M=2$ and $PY=1$ so you will have $V=\frac{1}{2}$.

Furthermore, note the equation for velocity actually calls for nominal output not GDP. That is the equation is given by $V=\frac{PY}{M}$, nominal GDP is just gross nominal output (in fact even this is a derivation the relationship actually is $V=PT/M$ where $T$ is number of transactions (see Mankiw Macroeconomics 8ed pp 108) but under very light assumptions one can derive that $PT=PY$ so we can run with it). This does not mean you can't use gross nominal output as a proxy, gross nominal output tries to capture $PY$, but you should be aware that there will be measurement error and GDP excludes gray/black market activity and so forth.

how much significance would the act of crossing below one have?

There would be no special significance to that per se. This would be like asking what would be special significance of CPI index crossing some arbitrary threshold like 1.05.

Optionally, what if V was negative?

$V$ proper cannot be negative because the smallest amount of times you can use unit of currency is 0 times. Its like asking if number of pies that you eat can be negative. That is simply impossible. This being said you could probably measure negative $V$ if you use $GDP/M$ to measure it but then you are running into a problems that arise from a fact that GDP is not exactly the same as nominal output.

For example, consider some state that completely disintegrated into Hobbesian anarchy, where all activity is done in grey/black markets with exception of 1 importing company, yet for some reason there are still statisticians going around and recording output in formal sector. Now let us assume that people get 1000 nominal income earned at grey and black markets, use half of that to import goods and services. Since GDP only tracks legal activity and since imports count negatively toward $GDP=c+i+g+(x-m)$ one would record negative nominal GDP and then $V$ will be negative as well, but that's rather due to limitations of measurement technique that you use not because $V$ as a concept can be negative.

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