## Wednesday, September 11, 2013

### The unit of account effect and the number base

Here and here I refer to the "unit of account effect" and I think the properties of the logarithm may help illuminate what this means. The unit of account effect follows from the prescription:
$$\kappa \rightarrow \kappa (NGDP, MB) \equiv \frac{\log MB/c_0}{\log NGDP/c_0}$$
that derives from the definition of the information transfer index $\kappa$ in the information transfer model. The effect itself arises because $\kappa \rightarrow 1$ meaning that $\partial P /\partial MB \sim 0$. Using the properties of the logarithm under a change of number base, we can rewrite this as:
$$1/\kappa = \log_{MB/c_0} NGDP/c_0$$
where $\log_{b} x$ means the logarithm of $x$ in base $b$. This means the unit of account property of money is analogous to the number base. (There is a happy coincidence of number base and monetary base terminology as well.) And we can easily see that as $MB \rightarrow NGDP$ we have $\kappa \rightarrow 1$. Basically, $x$ has one digit in base $x$.

Using values for 2013 (and $c_0 \simeq 11$), the (inverse) information transfer index is given by
$$\log_{MB/c_0} NGDP/c_0 \simeq \log_{283} 1443$$
Is the idea that the macroeconomy is communicating information via the price level in base 283 right now useful? Probably not in itself [1]; however, the idea the the monetary base represents the base of the number system used by the economy to transfer information is an interesting way to think about the unit of account in economics.

[1] Also not useful: $NGDP \sim 16$ trillion dollars in its "natural base" is $\text{5U}_{283}$.