Money money money

Nick Rowe did a recent post on his alpha and beta money (this time in terms of modeling Tony Yates "fairy tale" about escaping the zero bound) that he has discussed before. For whatever reason, when I read it this time, I thought about two different kinds of money in information equilibrium with each other and set about building the model:

$$
N \rightleftarrows M2 \rightleftarrows M0 \rightleftarrows S
$$

with $N$ being nominal output. Here M2 is Rowe's beta money and M0 (monetary base without reserves) is alpha money. However, I didn't take this much farther than that -- at least with regard to Rowe's model. This was just more for my own personal illumination. This information equilibrium relationship looks like:

$$
P_{1} P_{2} P_{3} = \frac{dN}{dM2}  \frac{dM2}{dM0}  \frac{dM0}{dS} = k_{1} k_{2}k_{3} \frac{N}{M2} \frac{M2}{M0} \frac{M0}{S}
$$

Use of the (slowly) time varying $k_{1}$ and $k_{3}$ can turn the nominal output and aggregate supply equations into identities by proper choice of the parameters because

$$
a^{\log b/\log a} = b
$$
where the exponent is our "model" for the information transfer index
$$
k_{1}(N, M2) = \frac{\log N/c_{1}}{\log M2/c_{1}}
$$

This allows us to solve the first information equilibrium relationship $N \rightleftarrows M2$

$$
P_{1} = \frac{dN}{dM2}  = k_{1}  \frac{N}{M2}
$$
$$
\log N \sim k_{1}(N, M2) \log M2 \propto \log N
$$
and see
$$
\log P \sim (k_{1} - 1) \log M2 = \log N - \log M2 = \log V_{M2}
$$

where $V_{M2}$ is the velocity of M2. The parameter choice is essentially $N_{ref} = M2_{ref}$. The interesting bit is where we solve for the best fit parameters for the middle relationship and set $P_{2} = CPI$. There we essentially have the old price level model except with M2 as the information source instead of $N = NGDP$. The model itself is pretty accurate:

Here is the information transfer index, graphed both ways I've defined it in the past (so pick the one you like):

For the front end, we have the nominal output model -- which is perfect, because we designed it that way (by choosing $N_{ref} = M2_{ref}$):

The information transfer index is approximately constant

And we can see that the price $P_{1}$ is proportional to the velocity of M2 (the y-axis label on this graph is wrong):

The model $N \rightleftarrows M2$ is basically contentless as I've put it -- it amounts to choosing the price $P_{1} = V_{M2} \equiv N/M2$. I repeat -- do not read anything into this pseudo-empirical success. It is a definition.

The only reason I put this piece here is to represent Rowe's alpha and beta money. One interesting take-away is that the abstract price in the AD-AS model $P : N  \rightleftarrows S$ is actually the product of the abstract prices in the individual pieces

$$
P = P_{1} P_{2} P_{3}
$$

Here we have (take $k_{3} = 1$ for simplicity meaning $P_{3}$ is constant):

$$
P \sim V_{M2} \times CPI
$$

meaning the AD-AS model that has the price level as the vertical axis is only valid for constant velocity.